Your description makes me think of Bioware games: the Mass Effect and Dragon Age franchises. Both explore multiple themes throughout each game, involve multiple cultures - the things you mention. Dragon Age in particular offers the player the opportunity to pick the culture they belong to. This different starting point is meaningful throughout the game (you'd see the world differently if you're a prince or if you're a member of the lowest caste, right?). I would very much recommend you play through both those franchises, to get an idea of what kind of stories games can tell.
A particular element Bioware's games are notable for is that the player gets to make choices, and those choices have an impact on the game. For example, you might get to pick whether to assist one faction get justified vengeance on another, or get the two to reconcile. Those choices have repercussions later.
In order to offer the player choices, you would need to look into branching-narrative. In essence, you would need to decide what choices the player has, and how each option would affect further gameplay.
Another element of storytelling in videogames is that the story is not necessarily told in a linear order. That is, the player might need to visit places A, B and C, but they can do so in whatever order they choose. For you as a storyteller, this means that A, B, and C are each a self-contained story, where the player would learn something and acquire something that's relevant to the overarching larger story.
To write that effectively, you would need to write stories within stories within stories, drop hints to one story inside another. A, B and C, while separate, would have to belong in the same world, be affected by the same overarching forces, elements in each might foreshadow the others, or assist in solving them.
Finally, consider the NPCs your player would be interacting with. Most videogames offer the PC "companions" - NPC friends. The PC interacts with those friends, gets to know them better. For each companion, you would have to have a story - a story that ties into the larger stories of the game, and that the player can help resolve. In addition to being "yet more stories for you to write", this is a powerful tool: the companions can organically introduce various elements of the world to the PC, and to the player. Also, the NPCs' stories offer the player a more personal stake in the conflicts of the game world. A threat to some random planet, for instance, feels considerably different from a threat to the home planet of an NPC who's been your PC's best friend for three games.
To sum up, think of a game not as one story, but as many stories that when put together, tell one bigger story. You need to be looking not only at the big picture, but at the smaller ones. A novel is not a good model for writing a videogame: first, it doesn't offer choices, branches that the player might take. In a novel, there's only one path forwards. Second, the narration of a novel is linear: chapter 1 is followed by chapter 2, which in turn is followed by chapter 3. A videogame would usually allow more flexibility, and you need to account for that.