I do call such things "mass formulas", but then again I am a number theorist, and one of my colleagues is a quadratic form theorist who specializes in such things. So this is mostly an expression of my specific mathematical culture.
I do not think that it is a standard term, at least not the only standard term. For instance, from another MO answer I noticed that some categorists call this the groupoid cardinality. This term in fact seems quite sensible to me, because the concept seems closely related to taking a quotient by the action of a group with nontrivial stabilizers and regarding the quotient set as a groupoid rather than a mere set.
As you say, combinatorially minded people speak of "Polya theory" or "counting with symmetry". Many algebraic geometers, upon seeing this phenomenon, would use the word "stacky". I wouldn't be surprised if there were other terms as well.
Overall I think this has the effect that a lot of people are partially rediscovering what is essentially the same concept. I would very much like to see a reasonably authoritative treatment of this subject appealing to mathematicians from different fields. Of course, I also look forward to seeing (better!) answers to this question.
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