Name for a type of idiom with two things joined (like "raining cats and dogs", "bread and butter")

Solution 1:

The generic term collocation includes idiomatically paired nouns, which both OP's examples are. So I'd say they are collocated pairs.

Other idiomatic usages such as strong tea and powerful computers are also collocations. In those cases, competent speakers recognise powerful tea and strong computers as "wrong", primarily because we're used to the other versions. Grammar and meaning aren't really relevant - it's just a matter of what people happen to say.

It's also worth noting that we never say it's raining dogs and cats. But as this NGram shows, we actually use that sequence more often than cats and dogs when we're not talking about the weather. Word sequence is one of the attributes of a collocation, though - coach and horses only occurs in that order.

Solution 2:

I believe the word for this is hendiadys: literally "one from two". The term is used when two things are joined together to refer to one.