The Sanc Kingdom in Gundam Wing (princess: Relena Peacecraft).
Gundam SEED has Orb Union (princess: Cagalli Yula Athha).
And Kingdom of Scandinavia as well (princess: Lacus Clyne. not really, but close — Clyne family is related to Scandinavian royalty)
Gundam 00 has Azadistan (princess: Marina Ismail), a very thinly veiled Expy of Iran, which was The Kingdom in Real Life until fairly recently. Azadistan is a deconstruction, since it's really anything but peaceful or homogenous. They conquered Kurdistan years ago, look down on Kurds or outsiders of really any type, are home to a boatload of religious extremists, and it's garnered the name Povertistan for a reason. The country soon descends into civil war, with the idolized princess being quite ineffectual, and has to be saved by the Gundams.
In fact, the original Mobile Suit Gundam's Republic of Zeon started off sort-of Kingdomish (Princess: Artesia Sum Deikun). Once it was taken over and renamed into a Principality by Zeon Sum Deikun's Evil Chancellor and his family, though, it moved rapidly towards The Empire.
Fleed from UFO Robo Grendizer -one of the Mazinger Z series- fits into this, in spite of being the entire planet. Benevolent monarch? Checked. A prince who is also The Hero (Duke Fleed) and a Princess (Maria Grace Fleed)? Checked. Princess has a romantic relationship with one of the heroes Kouji Kabuto? Checked. Ghibli Hills surrounding the Shining City? Checked. Invaded -and conquered- by The Empire? Checked.
Lupin III: The franchise occasionally uses this trope, but none are as easily described as Miyazaki's first feature film.
The Castle of Cagliostro was a dutchy; a small kingdom ruled by a Duke. The duke and his wife died in a large fire, while his daughter was at a religious convent. The Count was in a different castle, and now rules as regent. He plans on marrying Clarisse to become regent-for-life and discover the secret to the Cagliostro kingdom.
Fanelia and, to a lesser extent, Asturia in Vision of Escaflowne. No princesses in Fanelia, sadly, but much of the action of the series involved the king fighting off The Empire so he can take the throne. Asturia is a pawn of The Empire under the old king, but has a plethora of princesses, and as soon as one gets married and a clear line of succession is restored, they pull a Heel-Face Turn. Despite being a Duchy, Freid might almost count, although things don't turn out too well for them; the new ruler is not the heir by blood right and they wind up conquered by and ultimately cooperating with The Empire.
The Kingdom of Tristain in Zero no Tsukaima. To be fair, most of the other nations in the setting are kingdoms as well, but Tristain is TheKingdom.
Wakanda, the African Utopia in the Marvel Universe. Subverted with Latveria, which is not good. Though its run rather efficiently and pseudo-benevolently. And Doom referr to himself as "Emperor" every now and then, indicating that the kingdom of Latveria is really the (small) empire of Latveria.
Druidia from Spaceballs (Princess: Vespa) is also a planet kingdom, and one with its own enclosed atmosphere.
The Danish Olsen Banden movies tend to take place in such a version of Denmark, with the titular gang of bumbling sympathetic petty criminals often foiling the plans of villainous international capitalists, thereby saving the entire country.
In Scanners, the Mega Corp. ConSec fills this role, recruiting the heroic drifter Cameron Vale to neutralize the diabolical saboteur who has sworn a vendetta against them. Making them a weapons corporation is part of the film's elaborate plot to make us not really care about the stakes; the movie aims less for mere entertainment and more for making you uncomfortable.
Arendelle, inspired by Norway, from Frozen falls under this category. Queen Elsa accidentally thrusts Arendelle into an "Eternal Winter". At the end, she manages to unthaw the Kingdom.
The Star Kingdom of Manticore from David Weber's Honor Harrington series. The Applied Phlebotinum that makes this work is the Manticore Wormhole Junction, which allows the Kingdom to become massively wealthy through the sheer volume of trade the Junction supports.
In the recent books, though, it got subtly subverted by the fact that first the Star Kingdom of Manticore itself finally discovers joys of imperialism (Word of God has it that its name is now the Star ''Empire'' of Manticore), and second that the previous Empires are now either more or less good guys, or fair game for them.
The Old Kingdom in Garth Nix's Old Kingdom books. It has no other name, which fits as it is magically kept in Medieval Stasis in contrast to its neighbor to the South. And if the rightful rulers are not present, the Kingdom will fall into disrepair.
Played a little straighter (though not much) with the kingdom of Lancre, which is highly magical, presided over (currently) by a good and just king and queen, and actually ruled by a rather democratic mob of witches (who are in turn bossed about by Granny Weatherwax).
Well, the royalty in Lancre don't actually do anything; their job is to look official, the peoples' jobs are to politely ignore the royalty and get on with whatever they happen to do with themselves, and the thing in charge is actually the land itsself.
Except for the bit about a Queen in charge being a bad omen ...
In the story "The Eternal Champion" by Michael Moorcock, Ekrose is summoned to help the human kingdom of Necranal, which is at war with the Eldren. At first, it is played straight, but as the story progresses, Ekrose finds this trope becoming subverted with the human kingdom, with Fantastic Racism having a much larger role in the war than expected. It turns out to be played straight with the Eldren kingdom.
The land of Osten Ard in Memory, Sorrow, and Thorn has several political regions based on various Fantasy Counterpart Cultures, but the most typical of this trope is Erkynland, modeled strongly on Medieval England and home of The Hero, Simon. It rose to domination of Osten Ard on the strength of King Prester John's prowess in battle and remains there through the present time of the story. In contrast with the typical use of the trope, the majority of human nations in Osten Ard are mundane, but they are built on a land steeped in magical powers that humans only vaguely comprehend.
This trope is almost certainly why the area governed by Big Bad Galbatorix in the Eragon books rules an empire.
Mercedes Lackey's Tales of the Five Hundred Kingdoms are, well... fractured fairy-tales in exactly that setting, with enough variety built in that there will always be a beautiful princess, a tyrannical king, a scheming chancellor, etc. available whenever The Tradition requires one.
In the Star Wars Expanded Universe, there's the Hapes Consortium, an alliance of several dozen worlds ruled by a hereditary monarchy. Also, since it's a matriarchal society, the ruler is always a queen.
Gondor is a kingdom-in-exile. The line of kings there die out about 1000 years before The Hobbit, and since then has been ruled by the Stewards, making it a combination of this and The Federation. The good news is is that the last Princess of Gondor married the last King of Arnor, their fellow Successor State to the Kingdom of Númenor. It is from the latter line that the Rangers were established. The title of the last book indicateshow Gondor turns out. Rohan is also a Kingdom of sorts, and what parts of this Trope Gondor lacks, Rohan makesup for. Also, kingdoms are everywhere in the prequel The Silmarillion.
Barrayar in the Vorkosigan Saga calls itself an empire. However, it is only composed of three planets, and it is in many ways closer to this then to being an empire. It has a fairly compact government and is only a medium-sized state by galactic standards, lacking the sprawl characteristic of both The Empire and The Federation, as well as the spectacular tyrannies of many variations of The Empire. Moreover while it has a Secret Police and a Deadly Decadent Court, both of these are rather toned down and are far less sinister than many variations of this trope; at least they are since Aral's Regency.
Live Action TV
Merlin Camelot is essentially good, though the anti-magic stance makes some people feel differently. Arthur will obviously eventually make it more like the usual 'good' Camelot in time. There were aversions as well, Cenred's land for a while, for example. He only cared about himself and couldn't be bothered with the villages.
Plus, of course, the Royal Air Force. The Bretonnians have easily the best aerial troops in the game.
Warhammer 40,000 has an unusual example - the realm of Ultramar, eight systems ruled by the Ultramarines chapter of SpaceMarines. Its worlds manage to be both technologically-advanced and prosperous without turning into industrial hells, and under the benevolent leadership of the Ultramarines the citizenry is so happy that Ultramar is visible as a bright constellation of positive energy in the Warp. The kicker is that Ultramar is part of the setting's Empire, the Imperium of Man, a brutal totalitarian state.
The Kingdom of Aldis in Blue Rose is an excellent example of this — it's got all the trappings of a medieval fantasy kingdom, but magic and Magitek are sufficiently advanced and widespread that it's at least as nice a place to live as a modern first-world country, with excellent medicine, sanitation and communications. They've even got legal and accepted same-sex marriage.
Trappings is kind of the right word. The monarchy isn't hereditary, but is appointed by what amounts to an angel. Decisions are made via a small council where the monarch has 2 votes. The "nobility" is actually a civil service mandarinate, entered via egalitarian testing (including a morality test).
Obviously seen in Dungeons & Dragons, though "pure" cases may be a bit more rare than one would think — official D&D settings tend to feature quite an oddball mix of cultures, and this variety extends to forms of government as well. Still, quite archetypical canonical examples like Cormyr in the Forgotten Realms or the Kingdom (nee Grand Duchy) of Karameikos on Mystara definitely exist.
The Dukedom of Riskent in Super Robot Wars Original Generation (princess: Shine Hausen).
Ferelden in Dragon Age: Origins is mostly good (until Loghain takes over, anyway), just somewhat ineffectual.
Neverwinter is technically a city-state and ruled by a Lord (who happens to lack daughters), but otherwise fits this trope perfectly.
Final Fantasy VI's Figaro probably is the most remarkable. Figaro Castle itself is in fact a machine that can travel underground between the two deserts on the continent, and its King, Edgar, is one of the heroes that resist the evil Empire. It doesn't have a Princess though, much to Edgar's dismay.
Final Fantasy XII invokes this trope to the letter. The Archadian empire invades the smaller, peaceful kingdom of Dalmasca, which even sports a beautiful princess.
This article reads like a description of the kingdom of Hyrule (princess: Zelda) from The Legend of Zelda series. Hyrule seems to be working more like a a loose confederacy that a centralized kingdom, but the general theme is the same.
The eponymous setting of The Neverhood is a bare-bones version of The Kingdom, with very few residents besides the king himself.
Mostly because he was forced into hibernation before he could finish populating it.
The Mushroom Kingdom from Super Mario Bros. is an obvious example. It has Princess Peach, though all other officials are strangely absent...
Wyndia counts as this (to greater or lesser extent) in every Breath of Fire game it appears in (all but the fifth), and in the fourth also is part of The Federation against The Empire (which respectively fit these tropes). There is some minor subversion of this in the second, though, involving the Obligatory Princess.
Fantasinia and Bronquia (probably Embellia too) in Yggdra Union - except that the princess, Yggdra, is The Hero instead of the love interest.
The Pharastia Kingdom of Vanguard Bandits is cleanly this in the Kingdom Branch of the game. Then Subverted on the Empire Branch, as it becomes clear that the Kingdom isn't fully on the up-and-up either. A harsh lesson for the hero to learn, after a lifetime of idealizing it.
Despite being technically a Principality, Gallia in Valkyria Chronicles counts. The ruling dynasty, especially its Archduchess is loved by the people. It's another story, however, when it comes with both the bureaucracy and aristocracy in general.
Rakios in Eien no Aselia. The king is probably the least sympathetic character in the game after Soma, but Lesteena is nice. And just before she moves to assassinate him to save the country, someone else does it for her.
The Kingdom of Boron in the X-Universe qualifies, though its government is technically a constitutional monarchy akin to modern-day Great Britain (i.e. Queen Atreus is a figurehead for an elected government). They're portrayed as good guys, are constantly under threat from the SplitDynastynote though the Boron alliance with the ArgonFederation keeps them at bay somewhat, and have the smallest territory of the Commonwealth races. They've also got a few unique pieces of Applied Phlebotinum, such as ion weapons.
Battle for Wesnoth has a good kingdom that is turned to evil when the Queen has the King murdered during a war. The resistance centers around the last legitimate heir (or so he thinks) to the king that managed to escape assasination. The Princess on the other hand is misguided by her mother, and only a Enemy Mine situation gets her on the right side eventually.
Trent and Mercia fill this role during the "Storm Breaker Saga" of Sluggy Freelance, though they're treated a little more cynically than most.
The Tower in Tower of God. Ruled by King Zahard, it gets by pretty well, however, there is a reason why the heroes are Sticking It To The Man: The inner section of the Tower is a giant Thunderdome. Being tested on each floor, people are trying to reach the top to fulfill their greatest wishes, often harming each other. Usually, thousands of people give up, get injured, fall into despair or die for every person that reaches the top. Rules of power conservation tend to tear apart families and the Ten Great Families aren't really nice to their relatives. The Kingdom is not a very nice place to live in anymore and people want to change that.
Equestria fits the description quite nicely, even though technically (as it's ruled exclusively by Princesses) it's a Principality rather than a proper Kingdom.
The same with the Crystal Empire, as it's only The Empire in name only (it's tiny and - at least in recent times - good, governed by a Princess and formerly ruled by a King).
The Northern Water Tribe from Avatar: The Last Airbender fits this trope better than the actual Earth Kingdom (which is more like The Federation), especially since it comes under siege by The Empire. It also has a princess, Yue. The three Water Tribes (including the Foggy Swamp Tribe) only have one evil character (Hama) in canon.
The Earth Kingdom does fit a little, in that Long Feng had been keeping the king oblivious to everything, and had been doing sort-of-evil things in his name.
Animaniacs: In "King Yakko" Yakko became king of a country so small a magnifying glass was needed to see it on the map, whose economy was based on production of anvils. And then they were invaded by a neighboring dictator who wanted their national product, which the royal house of Warner was only too glad to give him.
As far as kingdoms to rescue, there's one called 'Loathing' that could use an adventurer or two. No official princesses, though. There's a rebellious one that hangs around, but she's not said to be part of the actual political hierarchy.
The Kingdoms within the British/Restored Empire from Decades of Darkness fits this trope more or less.
The City of Axiflos in Open Blue is a neutral city-state that was forced to defend itself from larger neighbors whoo tried to force it to join their sides. Suffice to say, it can't really be considered a Distressed Damsel, as centuries of defensive buildup has turned it into a Stone Wall that not even the largest empires dare to invade nowadays. Such was its reputation that the Axifloan Coalition was named after it.
Most real-world kingdoms and small countries see themselves as The Kingdom, especially the governments themselves.
Most constitutional monarchies see themselves this way, especially the ones in Europe and the Commonwealth. This is especially true since the two main powers of the Cold War were far bigger federal unions, the USA and the USSR.
Kingdoms in Real Life can differ much. There were kings like Louis XIV of France who reigned absolutely and could say "L Etat Cest Moi" (well okay, not really, but let's just consider 'absolutist' rulers absolutist for the sake of simplicity, m'kay?), medieval kingdoms where the power of the king was limited by his vassals (Ur Example for the anglosphere could well be the way the barons of England forced King John to sign the Magna Carta), and nowadays we have many representative monarchies which are essentially republics with a crowned head of state. In the past, there were even kingdoms where the king was elected! (the workings of which paralleled the 'noble republics' of Italy where a small group of wealthy families monopolized power, the only difference being councils of nobles versus councils of wealthy merchants and freemen voting for the top dog. And a certain amount of pomp and grandeur).