Blue's rematch team includes a Pidgeot with Return, which is always at maximum power when used by the AI. Checking his team in the original games, a Pidgeot is on his team (in varying evolutionary stages) in every battle but the one in the lab. It's probably the same Pidgeot, which he's had for three years — it makes sense that it would like him a whole ton!
The Teddiursa line being exclusive to Gold and Phanpy line being exclusive to Silver in the North American editions of the games makes more sense if you look at their respective palettes.
In HeartGold and SoulSilver, when you visit Copycat's new home, her parents mention that she lost the doll a boy gave to her a few years ago. Go upstairs and visit Copycat and what Pokémon does she have? Banette, the abandoned doll Pokémon.
Petrel's team had a more varied 3-Mon team in the first battle but a 6-Mon team with five Koffing and one Weezing in the second battle. This fits his incompetent personality perfectly — of course he's going to overload in numbers instead of creating a more balanced team when he probably follows TR's philosophy to the letter rather than working intricately!
In Crystal, a Pokéfan on the S.S. Aqua has a rather pathetic team of a Furret and four Sentret. After the battle, she remembers that she has to get her Pokemon out of daycare. How'd she wind up with such a weak, repetitive team? Her other Pokémon is a Furret of the opposite gender, and when she kept leaving the two Furret in daycare together, she wound up with a bunch of Sentret eggs. This could also explain countless other trainers with four or five identical Pokémon. At least it sounds better than them catching the same thing five times over out of stupidity.
Red's team is pretty much based off of Pokémon that were canonically encountered by or given to the player character in the Gen I games. The starters (counting Pikachu) are given by Professor Oak (or by different NPCs scattered around the region, if you're playing Yellow); Snorlax was awoken and captured either west or east of Celadon City; Espeon (in the originals) was found in the Celadon Mansion (as an Eevee); and Lapras (in the remakes) was given by a Silph Co. employee as a thank you for stopping Team Rocket.
From the game player's perspective, there's little in the way of replay value or even purpose in the game after the Elite Four/Blue are defeated, the major options left are to collect all of the Pokémon or raise your team to absurdly high levels. By yourself, it's impossible to complete the Pokédex without trading with other people, so that just leaves endless level grinding. This means that, by isolating himself at Mt. Silver, Red has, in a meta-sort-of-way, done exactly what any lonely player too attached to playing RGBY would eventually do with nothing else to do. Players still on RGBY when GS were announced might probably still be level grinding their team to hold out until the new game. So what's Red doing on Mt. Silver? It's not like he had anyone else to talk to and there was nothing else for him in Kanto as the Champion, and the Pokémon in Mt. Silver are even stronger for level grinding than those in Cerulean Cave, so it's like, why not? Mt. Silver is also in close proximity to Johto, so maybe Red was hoping he would be discovered by a different Champion from a different area for a challenge. By GSC, the fruits of Red's labor are finally witnessed, with three years worth of isolated level grinding unleashed into trainer battle at last. Red must have been (silently) ecstatic to have a new purpose when Ethan/Kris stumbled upon him for battle, as this gave Red something new and productive to do for the first time in three years. Red leaves the mountain after being defeated because he was satisfied that at long last there exists another trainer stronger than him, finally moving on to do other things.
There's also the possibility that the Red in Mt. Silver is a ghost, and that by defeating him you finally put his spirit at rest, and he can ascend to a higher existence. It's the way his mother in Pallet Town refers to him. She says stuff like "he's been gone a while", "I'm worried about him", "he'll be back soon", etc. She's probably in denial that her son is dead. Blue's nature also seems different in GSC than it does in RGBY, so it's also possible the realization that his rival had passed on led to Blue no longer having an equal in strength, and thus fell out of interest in battling anyone, which is why he is less arrogant and impish in the second generation (if you choose to disregard the Raticate theory, that is).
Red's mom is worried because that's what a mom does when her son is out having adventures. Her very first dialogue consists of her saying that she'll miss Red when he's out traveling. Blue's personality is a little better in Gold / Silver simply because he's older now (he wasn't malicious in Red / Blue, just a brat). Plus, he doesn't have any reason to antagonize the player character anyway, as he's not his "rival".
Red and Blue reappear in Pokémon Sun and Moon, and have both visibly aged, meaning Red is still alive and well.
Why is the Daycare Lady in HeartGold and SoulSilver so convinced that you and the opposite sex friend are a romantic couple? Because she's used to thinking about sexual compatibility and pairings, given that the daycare is also often used for breeding Pokemon, and she probably got used to being around Pokemon who are sexually active.
Also, the Daycare Couple are the grandparents of the player's best friend and neighbour. It isn't uncommon to have close relatives who ship you and your best mate, especially if the two of you grew up together.
At first, Silver (the rival character) follows the standard Team Rocket philosophy that Pokémon are nothing more than tools. However, he starts to question this after being defeated by Lance, and he mentions on Victory Road that he's starting to understand what the player has that he lacks. His character arc is shown to be complete when you fight him for the last time at Indigo Plateau (and, in the remake, if you have a tag battle with him in the Dragon's Den): his Golbat has evolved into a Crobat, only possible at full happiness.
The rival turning over a new leaf is made clearer in the remakes. In it, he offers to return the stolen Pokémon to Professor Elm, only for Elm to let him keep it after seeing the fondness the Pokémon has for him.
There may be a reason that Whitney and Clair both refuse to hand over your badge at first — think about what they have in common, and think about how many tries most players have to make to beat either of them. Odds are, neither of them are even used to losing. (With that in mind, it makes Sabrina look surprisingly mature as well).
Clair's dialogue—specifically, that she can hold her own against the Elite 4—goes even further. She implies she has literally never lost to anyone other than Lance before. No wonder she gets so touchy at her defeat!
In RGBY, you get to capture a Snorlax. There are two Snorlax in RGBY. In GSC, when you go to the Kanto region, the setting of the previous game, you can catch also catch a Snorlax, but there is only one. Because Red, the protagonist of the previous generation, caught the other one.
The Pokéwalker, may seem like just another attempt by Game Freak to make nerds get some exercise. Remember, however, the Pokémon Pikachu 2, a step counter that was released around the same time as the original Gold and Silver. Not only did they reboot the games, they rebooted the accompanying gadget!
Also: you're walking around, hoping to get the points needed to encounter a Pokémon. Just like in the game, you probably have kept walking in tall grass, hoping to encounter a Pokémon. Oddest De Fictionalization ever?
The change in gender determination listed under Game-Favored Gender (based on behavior, not physical strength) reflects how for some species, the females would be larger and physically stronger on average, such as Ariados. It's simpler to standardize sex determination as independent from the Attack stat than to have a different formula for each species.
Why did Route 23 just disappear, even in the remakes? Remember those wild Larvitar very close by in Mt. Silver? They have to eat a whole mountain to grow.
The final Rocket Grunt in Gold/Silver/Crystal, who spoke in broken English in the English-speaking versions, apparently spouted Gratuitous English in the original Japanese, and in HeartGold/SoulSilver, it's revealed that he is indeed a foreigner and that he intends to return to his homeland to try to restart Team Rocket there. He's then found in Icirrus City in Black & White. Whereas the first four regions were all based on regions of Japan, Unova is based on a region of America — presumably their native language really is English whereas the native language of Kanto/Johto/Hoenn/Sinnoh is Japanese!
And when you keep in mind that all of the other regions were based off Japan, and Unova was the first to be based on America, when he says he will try to restart Team Rocket in his homeland (presumably Japan), he is actually referring to the fact that in every Japanese region, Team Rocket (or whoever happened to be there), was wiped out and shut down.
Red doesn't have Espeon in HGSS because back during the FRLG era, he wouldn't have been able to evolve his Eevee into that particular Pokémon, even though he has obtained the National Dex and is able to catch Pokémon that are not part of the original 150.
Though the Dark Type was touted under one of the new features of these games, you get only Umbreon and Sneasel before reaching Kanto... and you can only get Sneasel before the player can get to the 8th Gym! Though several non-Dark Pokémon can use Dark-type attacks, this would be a problem if you want to take advantage of STAB or the immunity to Psychic-type attacks. The Ghost-type Ecruteak Gym and the Psychic-type Will of the Elite Four account this lack. More specifically, all of the Pokémon in the Ecruteak Gym are part of the Gastly line, which are half-Poison; the pure Ghost-type Misdreavus, despite being a Generation II Pokémon, is absent. Someone can either use one of the several Normal-type Pokémon that can use Dark-type attacks (and are immune from Ghost-type attacks) or use a Ground-type Pokémon where Ground-type attacks are super-effective. note In Generation II, Abilities, including the Levitate Ability, did not exist. This principle carries out to Will: all of his Pokémon are dual-typed. You only get some more Dark types in the location near the Psychic-type Saffron Gym, which uses a variety of Psychic Pokémon. However, this leaves the question on why some Generation II Pokémon are not located in the Generation II region Johto but rather in the Generation II region Kanto.
While Karen only having 3 Dark-type Pokémon despite being a Dark-type specialist could be explained as there only being a small number of Dark-types available in-game, it also makes sense in-universe: as Karen herself tells you upon defeat, her belief is that trainers should try to win with the Pokémon they love most! Karen may have a preference for Dark-type Pokémon, but she has Vileplume and Gengar on her team because she likes them and is following her philosophy of winning with the Pokémon she loves best!
The Blackthorn City Gym has lava all over the place. Seems kind of weird for a dragon gym, but then you remember that ice is their only other weakness.