pretty sure that everyone before me already replied succinctly to your questions but I figured giving my two centavos on the situation wouldn't hurt. (Also cause I'm drafting an Isekai Novel rn and thought this thread was reading my mind haha.)
For me, it's the situation of the MC being in a new world completely separate from their own. (I also like MCs who bring 'our' world knowledge into the other worlds'.
Usually, MCs hate their original world, that's the main cliche of an Isekai though there are some others where they do want to go back home and that's the main drive of the story. Like The Devil is a Part-timer.
In most cases they end up loving the new world and want to stay.
I know that you're looking for Video Game related answers but I'm not into JRPG Isekai Games (and the few western ones out there) so I'll base my answers on anime/manga that feature Video Games and Gaming as the focus of my answers and references.
Are you going for a self-aware Isekai? Like what SAO (anime) did with one of their 'NPC' character? Most Video games that I know of that have a 'breaking the 4th wall' concept aren't Isekai but really it just depends on whether you want to be the 'Scream' of Isekai - A.K.A metacommentary of Isekai or just use Isekai tropes in a story.
I remember reading a Taiwanese Manhua, the title was 1/2 Prince I think and they had the concept of NPCs being real-life characters who had their memory wiped/dead/comatose. Though I might be misremembering that. There's also Overlord where the NPCs are created by other characters and their backstories are from their creators.
For an actual Isekai game, there is Genshin Impact, however, the MC you play as doesn't reveal any backstory about themselves which -granted can make sense due to the MC being technically the player (a.k.a. us) We can switch between other characters in-game but 'canonically' we are playing/fighting/exploring as the MC and their companion only.