The history of Games Workshop games does not begin with Warhammer, but to a large extent, the company that it is now began with Warhammer. Prior to the release of Warhammer 1st edition, most fantasy miniatures were used as props for Roleplaying games (which themselves evolved from Wargaming, but that's another story). Warhammer was essentially a slightly different thing to do with your fantasy miniatures. I put "slightly" because 1st edition Warhammer was still substantially a Roleplaying game. It was supposed to use a Games Master, have player created characters and "adventures" rather than scenarios. The concept of competing players each with their own armies, really developed from 2nd edition onwards.
The 1st edition Warhammer rule books including character generation
However, the roleplaying concepts were not forgotten. 3rd edition Warhammer is particularly interesting. The 3rd edition rulebook was written as if it were a stand-alone system, the dedicated army list book Warhammer Armies, came later and appears to have been written to support tournament play. The basic rule book has no army lists, but it does have an extensive bestiary of fantasy creatures with points values for all of them. In addition, it includes rules and points values for a number of specialist troop types. These included Shock and Missile elites who were simply better at fighting or shooting, but also included more unusual troops like scouts, skirmishers, assassins, flagellants and berserkers. The interesting thing is that these specialist types were included as upgrades to existing troops. So if you wanted to create a goblin assassin or an elf berserker you could, and adjust the points values accordingly.
Characters were also presented as upgrades. They were given levels from 5 to 25 and these represented the number of bonuses applied to troops' profiles. The interesting thing was that these bonuses were variable, with six different options presented allowing more combat orientated or leadership orientated characters (though one was presented as the standard version to use in tournaments).
When Warhammer 40,000 was released a similar system was presented, with tables for randomly generating troops and characters and with an emphasis on scenario-based game play.
In subsequent editions, these ideas have been diluted, but elements remain. Standard Games Workshop characters are still generic types upgraded by players with equipment and/or magic items. Units vary in numbers and can frequently be upgraded with different weapon options as well as extra elements like standard bearers, musicians and champions.
This sort of approach is common for games produced independent of a range of models, such as Hordes of the Things, Armies or Arcana, Song of Blades and Heroes or Warrior Heroes: Armies and Adventurers. This is understandable as the game needs to be adaptable to players miniatures. It is more unusual for games and miniature ranges produced together.
For example in War Machine and Hordes, the number of models in a unit can vary, but not their equipment. Some can be upgraded with additional specialist models, but not all. Warjacks and Warbeasts are defined types with standard equipment. And all characters are named individuals with a defined range of abilities and equipment. The same is true of Malifaux, Helldorado, Bushido and Anima Tactics. All of them emphasise named characters with only limited, if any, upgrade options for individual troops.
Warhammer's origins as a hybrid roleplaying game has bled across into other Games Workshop games. Inquisitor, with its player-created warbands of characters has already been mentioned, but it also present in Mordheim, Necromunda and Blood Bowl. All of which feature a campaign system that emphasises gangs/warbands/teams developing over time as games are played. This system was transferred to the Warhammer Historicals Legends of the Old West and Legends of the High Seas, though the first time something like this was attempted was in Realms of Chaos.
Interestingly, it is not an approach adopted by most other Fantasy and Sci-Fi skirmish games. Bushido, Infinity, Anima Tactics, Malifaux, all of these games are of a scale similar to the Games Workshop games, but none of them have taken this route, instead opting to focus on characters created by the games designers, rather than the players.
Most of these games focus on a strongly defined game world, many with an evolving narrative. This is particularly true of Malifaux, whose narrative has developed with each rule book to the point where it seems that the writers aren't entirely sure if they're writing a rulebook or a novel. It is also true of War Machine and Hordes. In contrast, the narrative of the Warhammer world has hardly moved on in twenty five years. The game world has changed, but this has mostly consisted of retconning the existing background to fit the development of new armies. There is not a defined ongoing story.
And yet this actually fits better with the narrative approach that Games Workshop takes with its games. They present a well defined, but essentially static game world and invite players to create their own characters and develop their own narratives. The campaigns that have been built up around the Games Workshop skirmish games develop a life and a story of their own, rather than following a story defined by the game writers.
In contrast, most of the other games listed above present a strong, developing narrative and use it as a back drop. Players have essentially three choices. The first is to ignore the background and focus on the game. The narrative provides a theme to build a game and miniatures, but little more than that. This is essentially the way that card games, such as Magic the Gathering or board games work. Nobody thinks that Settlers of Catan accurately reflects the settlement of an island or Monopoly running a property empire, but it doesn't matter. The second option is to try and play within the limits of the background, much as some historical gamers prefer only to play games based around known historical events and campaigns. The third option is to develop your own background or play "what if" games that develop the official narrative in a direction of the players choosing.
What these games pointedly do not do, which most Games Workshop games do, is provide a narrative with sufficient space for players to develop their own characters and stories that still fit into the overarching story of the game world.
For some players this does not matter. Many players view wargaming as imply gaming and the narrative is of minor or no importance. But for others, developing a narrative around your characters and armies is part of the enjoyment of wargaming. And, for all the legitimate criticism aimed at it, this is one thing that Games Workshop has done well in the past and that other Sci-Fi and Fantasy games, for all their positive qualities, often neglect.