What you are describing is a concept of the mechanically bisexual. The options as given often allow players to choose in a sandbox game whether they experience the game as a completely non-queer experience or not. It sometimes creates queerness as an option rather than a core part of an experience which rep wise is considered a step better than when all romance options in games were mandatorily heterosexual but also kind of a cop out where player choice means all characters are often Shrodinger's bi. If you want to experience say Skyrim as an almost entirely queer free experience - you can. Your choices flip that representation on and off like a lightswitch so if you have queerphobic tendencies the game doth not offend much. No one ever hits on you first.
Rep wise Gay characters are ones specifically ones where the queerness isn't optional, it's a part of the canon of the character. Straight characters often are so in fixed story narratives where they have hetero relationships and if they have brushes that look like same sex romance it's played for laughs and treated as not really an option. Since culture still sort of assumes straightness as a default if the character only ever is coded romantically by the frame of the game to be attracted to the opposite sex they can be termed a "straight character" because as a player the game's interfacing with that character's sexuality is mandatory. An example is the Prince of Persia games or the Final Fantasy series which have a romantically coded opposite sex paramours that you don't have an option not to interface with the character's sexuallity.
This is way more common in older games and fixed story franchises.