#P-complete problems are at least as hard as NP-complete problems
It's not at all weird to compare the complexity of counting and decision problems: reductions allow you to do that very naturally. You can use reductions to compare any kind of problem: decision, counting, approximate counting, optimization, function problems, ... All of this follows from the intuitive notion that a reduction from A to B means "If I could solve&nbsp;B, I could also solve&nbsp;A at roughly the same cost."

The existence of easy decision problems whose counting version is hard is a red herring. Counting matchings is the hard problem: if you could do that, you could solve any NP problem by reduction. Detecting matchings is easy: you can do that but it doesn't (seem to) help you to solve NP-complete problems.