My inclination is to make them synonyms, if that works as you describe (people who tag a question as "counting" simply get it automatically re-tagged). At least right now it doesn't seem like the counting vs. noncounting facets of combinatorics need separation here, though that could change in the future.
I think my earlier comment was misunderstood. It does not currently seem clear that there is a reason to merge the tags or make them synonyms, at least not without someone with expertise saying this is reasonable. The point of my comment was that maybe we should retag the appropriate questions and not mess with the tags. Maybe there are people who know what combinatorics is and what counting is and mean counting, not combinatorics. Unless this impossible I see no reason to change the tags. Changing questions that are inappropriately tagged is something we will always have to do, and it isn't that big of a deal.
The point about AT and topology is a valid one, and until someone points out the harm of having multiple tags for things that are similar but not the same I think it wise to leave the tags unchanged.
At least 2 of the counting questions seem to be about counting and not combinatorics (maybe I am wrong, but the "why do our numbers start over..." and "what else can an elliptic integral count?" don't seem like combinatorics a priori.)