This trope is ubiquitous in tabletop roleplaying games. It is quite likely that, regardless of genre or conventions, you are making a character who belongs to a pattern or group or class which is found throughout the gameworld. There are two main ways to do this: classes (with or without levels) and fictional organizations, clans, nations, and tribes the players affiliate with. (Both are frequently used together.)
BattleTech: This is encouraged by the game's design. Major House regiments and Clan formations have been detailed in canon only in their basic composition (X number of mechs, Y number of Battle Armor, Z number of Aerospace Fighters, etc), allowing for easy insertion of a fan-created team. Also, Catalyst flat-out encourages players to make up their own new mercenary units, militia groups, pirates and other formations. This series has entire sourcebooks based on famous in-universe units and provides a metric ton of one-off short bios for various minor characters in the setting. Fans have alternately elevated certain characters to Memetic Badass status, put their own spin on minor characters, or written their own characters into existing units. Save for exceptionally out-of-place examples (no, no Inner Sphere mercenary unit aside from Wolf's Dragoons is known to have a Clan Omni before 3050) most fans tend to accept it at it comes. Alternatively, if you want to create your own Battlemech, there's entire design programs for that, as well as many stories of lost prototypes and one-off modifications to justify its inclusion in the setting.
The Dresden Files: In a crossover with literature, it seems like almost every group of the RPG has an Autumn Knight. Not spring, always Autumn. Additionally, in the series there are three main vampire courts, the Black Court, the Red Court, and the White Court. A Jade Court has been mentioned in passing, and Billy brings up the possibility of there being different colored courts.
The 2e Alchemicals splatbook makes room for not one, but two separate Sailor Earth Exalt types. Alchemicals are divided into castes based on the magical materials, each of which corresponds with another type of Exalted — e.g., Orichalcum with Solars, Moonsilver with Lunars, etcetera. However, the Adamantium caste has no equivalent, opening the possibility of Exalted with a natural affinity for that material, and the sourcebook states that, since Abyssal Exalted are a later corruption of Solars and were not intended to exist, the option is also left open for a new Exalt type who has a "true" affinity to Soulsteel. These two types are only mentioned as vague possibilities — nothing to suggest what they should be, if they exist. It was suggested in Dreams of the First Age that now-slain Incarnae existed once, and that their Exalted went mad and were wiped from existence; this is one potential explanation of the above. However, given the source and given that it is generally considered to be quite feasible — if far from easy — to take on one's almighty divine patron and win, the fan base has reacted to this idea with little approval.
On the Terrestrial side, while canonical Dragon-Blooded can only be aspected to the five Gaian elements (the Western four, plus wood), it's a common hobby to try to make Dragons of the five Underworld elements (ash, bone, pyre flame, blood and void) or the six Autochthonian elements (crystal, lightning, metal, steam, oil, smoke).
Also for the Abyssals, their sourcebook says that there are thirteen Deathlords, but only nine of them are described and given stats. This is explicitly so that Storytellers can create their own Deathlords.
Introduced in 3e, the Exigents are the chosen of minor gods (there being gods of everything in Creation) who prayed to the Unconquered Sun for the permission to make their own Exalted when their domain was threatened, and used the divine fire of the Exigence they got if the Unconquered Sun approved (or if they stole it somehow), with the Exigence magnifying the god's own power in the chosen. This takes a lot out of a god though — one sample Exigent got her powers from the god of a group of fields, who disappeared entirely after empowering her. Apparently the game designers were specifically invoking this when including them, as they wanted a way for players to introduce any Exalted they wanted without having to worry about adding a new group of the Princes of Earth into the game.
If you do want to introduce a new type of Exalted, 3e has you covered there as well, since it introduces new Exalted like the Liminals and Getimians alongside the established types. (Admittedly, the big obstacle for fans in introducing new Exalted is coming up with a new charmset for them, which's never been an easy task.)
3e also introduces three optional Exalted for fans to flesh out: the corrupted, mind-eating Hearteaters, the haunted Umbral Exalted, and the reality-warping Dream-Souled. Word of God is that the Hearteaters are the Adamant-affinity Exalted — it's not related to what they are, but to what they were before they fell.
The Spoken, an extinct 3e Exalted type that lived undersea, open up possibilities beyond that of their existence: firstly, they're the first known non-human Exalted, previous editions having limited all forms of Exaltation to humanity, and secondly they confirm it is indeed possible for an Exalt type to be eliminated (from what's known of them, they were created Exalted, so it'd be entirely possible for something to happen that wiped them out and prevented more from being created).
First Edition's Games of Divinity established at least 23 Primordials were imprisoned in Malfeas as Yozis, if not more, of which 15 have been recorded over the course of the game to date, leaving at least 8 for fan writeups. The number of Primordials who were killed and became Neverborn is unknown, leaving them up for grabs too. The number of Primordials who simply never participated in Creation, or left before the Primordial War, is also unknown. Then you have Second Edition's Infernal Charm Triumphant Howl of the Devil-Tiger, which allows the Infernal Exalted to become Primordials themselves. Basically, there are a lot of ways to introduce a new Primordial.
Linked to the above in Second Edition was coming up with Charm sets for known Yozis for Infernals to access. This doesn't apply for Third Edition's Infernals, as essentially their entire setup is different, and they don't require drawing up an entire Yozi's Charmset to have Charms like a particular Yozi's.
The Five Maidens are named for the classical planets (Mercury, Venus, Mars, Jupiter and Saturn), and some fans have taken the opportunity to postulate the existence of Maidens named for planets discovered later, helped by the books suggesting there might be additional Maidens.
While we're on the Maidens, one of the possibilities 2e's Sidereal splatbook proposed for the identity of the Convention on Oversight was the Maidens' imprisoned brother Nox, an Incarna on par with his sisters. He became the jumping-off point for a popular fansplat, the Nocturnal Exalted.
Magic: The Gathering: The Multiverse is a very big place, with a wide assortment of planes, races and cultures. As such, creating a Planeswalker, species, or even an entire plane that can fit into canon is possible. The only exception are that certain beings can't have a Spark, such as the Phyrexians, undead, or beings made of mana like Angels or Demons— and even that one has edge cases like Ob Nixilis (a Demon of Human Origin whose Spark ignited when he was still human), or Calix, a being created by the Therosian god of fate.
Old World of Darkness: The games usually includes some form of lost group for each major creature type. In Werewolf it's the White Howlers, Croatan, and Bunyip; Vampire has the Cappadocians] and in Ars Magica, something of a predecessor to Mage before its Dark Ages book came out, it was House Diedne, all mentioned but not present in the setting. It's a Sailor Earth player's field day. There's also the rules for creating new bloodlines in Vampire, the different "lesser" Mage factions, the Siberakh werewolves (Silver Fang/Wendigo hybrids) and all the other werebeast species (cats, snakes, spiders, etc.), and so forth.
Scion is a game where All Myths Are True, allowing fans to do their own writeups for gods and pantheons not covered in canon as potential Scion parents.
Traveller has systems to generate new sectors of worlds and design new starships. The game has also designated the Foreven sector, right next door to the popular Spinward Marches sector, as a "GM's Preserve". No official material will ever be published about it beyond the star positions and a few worlds with minor details. Game Masters are encouraged to make up whatever they want to occupy that space.
Warhammer 40,000: This is outright encouraged. Every major faction is divided into large numbers of subfactions with its own distinctive colors, traditions, quirks, and combat specializations. Canonically, only a minority of each of these is named and described, encouraging players to come up with their own personal factions when painting their armies. Among the Space Marines, for example, there are over a thousand Chapters, and only a hundred or so have been so much as mentioned in canon — it's even perfectly valid to imitate an existing army due to the way successor chapters are created from older ones. There are also two Primarchs whose identities and abilities are unknown, seemingly for this trope. Similarly, there are a million worlds in the Imperium that your homebrew Imperial Guard regiment could originate from, the Eldar are so secretive that a previously unheard-of Craftworld suddenly appearing wouldn't be out of place, and the Tau are always settling new Septs. And so forth for Dark Eldar Kabals, Tyranid Hive Fleets, Genestealer Cults, Ork tribes, Necron Tomb Worlds, Chaos Space Marine warbands... there's always far more of them than are ever actually detailed, and there's no shortage of players willing to make a neglected bit of background their own.
The Chaos Codex. Not only can you invent your own renegade chapter, you can also mix and match various legions and chapters into one army, with the justification that they are all united under one banner for some cause or another. This was originally the concept for the Red Corsairs, before they had their combat doctrine retconned.
8th and 9th edition codexes even include rules for creating custom faction bonuses, either inheriting existing buffs as a successor chapter/guard subdivion/et. al., or a properly-themed pick-and-mix to fit the bonuses to the player's new lore. 10th edition moved to a subfaction-agnostic system that made using custom armies even easier.
One of the biggest reasons why the 5th-6th edition Codex: Space Marines fluff by Matt Ward was hated was because it discouraged this, claiming that any chapter that wasn't just like the Ultramarines was bad.
Matt Ward also, ironically, gave the Necrons more character by retconning their backstory so that each individual Tomb World had more characterization to it rather than just having a different color scheme. Previously all were controlled by the C'tans and had little free will, meaning that, apart from the paintjob, they were more or less identical (although that was sort of the point).
