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For example, name some games you know, regardless of the main genre, where you could create your own character with a portrait and stats (and, let's say, a strategy game like Battlecry and an RPG like Baldur's Gate and Icewind Dale would also fit here).
Ple-Sen already mentioned NWN, but quality is more important than quantity in my opinion, so I will elaborate. NWN is an RPG from BioWare with a campaign that copies the plot of KotOR and two more expansions that have their own, slightly more interesting plot and narrative. The game is based on D&D, and character creation has the features that come with it. However, in my opinion, the most interesting thing about the game is the fan-made modules, which are independent stories simply written on the original game's engine. Their quality is sometimes truly outstanding, for example, the Prophet series has one of the best plots in video games in general and a stunning final dialogue in particular. I can list some other good modules, if you want