In Gaming.SE this type of question is very popular. Should we consider them on-topic?
Yes, that's a great kind of question for this site. We have an example already, where it's worked out very well. (Speaking of which, we should probably decide on one tag for both of these).
These questions have an objective answer (the original game the questioner was thinking about). They can have multiple potential answers, if the questioner wasn't specific enough, or some answers may provide more detail that would allow the questioner to positively identify the game in question. If there is not enough detail to answer at first, this can be clarified through comments and edits to the original question.
So yes, I think that this format works very well for StackExchange, and think that it's definitely on topic for this one.
I would suggest to always use the [identify-this-game] tag instead of the [Name-that-game] tag. It should be set as a tag synonym.
This tag name come from gaming.stackexchange (71 questions with this tag so far), so I think we should keep it as a standard.