Would anyone be against this?
Yes, many people, including me.
It makes sense to have these tags since there are plenty of artworks making a direct reference to the game, either by having the full cast present, or if a playable character is wearing an outfit from a given game (they are pretty distinct - Reimu for example). Other examples will include if a spell card is shown (it's rarely tagged but should be tagged, as it references the game), or if gameplay elements are present (ability cards from Unconnected Marketeers do count, but arguably animal spirits from WBaWC might not count unless they are presented as items). Some other examples that are applicable are if a pose is copied from the game (post #4495699 for example).
The problem is, it seems like it's often used when the character in the image originated in that specific game in the series but that is not tagged with any consistency
Most uploaders will slap on a game copyright tag if it shows up as a translated tag and not care about the standards to follow, because they don't care about Touhou or haven't read its wiki. If this is a problem, the solution is to simply garden these tags; add them to applicable posts, and remove them from irrelevant posts, and point a repeat offender to the wiki. In my opinion, I think the touhou wiki should be rewritten to better clarify when individual games should be tagged, as "making a direct reference" to a game can easily be misinterpreted by many uploaders/taggers.