I think it would make sense to specify both primary genres and secondary genres when specifying genres a game. For example, Sonic Adventure DX is primarily a platformer and action game, but there are also two pinball tables in the game. These are only in one stage, so it doesn't quite make sense to list pinball with the more pertinent genres. Of course, this gives some ambiguity: is SADX also a fishing game because that is the gameplay of one character (ditto for shooting)?
Separate main and minor genres for games
It is a good suggestion but my main concern is ambiguity. I think we either choose to list Sonic Adventure DX as a Platform game only, or we would be forced to include all secondary genres. Then you get into the silliness of listing The Elder Scrolls V: Skyrim as everything under the sun - role-playing game, survival horror, stealth, crafting, horse, vehicle. flying etc.
This taxonomy is just the first step in helping to categorise the games. The next phase will be to introduce a port history, and then other aspects like a paragraph on 'gameplay' where you could include description of the various bonus stages in SADX. I think it is relevant as people ought to know what kind of content is inside the game even if it's not 'primary'. For example my wife really struggled with the 1st 'FPS' section in the Walking Dead because she really struggles with motion sickness and lack of invert-y - even though 99% of that game has no shooting.