Like,
this seems very out-of-context. It'd be nice if they had link back to the group like threads about games do. Those that appear in the forum have that link, but not within the thread itself.
I don't know how it's done with games, but if tags have an ID number, then you could just add another field to threads that state which type of reference it has, and store the game/group ID as you do now.
Like: reference type: game/group, reference id: game ID / group ID
If, however, groups don't have id number, then this may be a bit more difficult to accomplish, but I'd suggest you assign them ID numbers instead of doing something more complex. The ID number needs only to exist for groups that have the group page, though.
The reference type could be enum or integer. Enum will be less easy to expand when seen necessary, but will catch errors if used wrong.
Usually it's good to have auto-incrementing unique ID in database tables anyway.
Done.
I don't know how it's done with games, but if tags have an ID number, then you could just add another field to threads that state which type of reference it has, and store the game/group ID as you do now
It was already like that. :D