The idiom can be considered both (1) to be extra-grammatical (here hijacking a reasonably common if far from simple sentence structure and using it in a strange way) and (2) to be using words in a strange way.
We'd expect a weather-it sentence to have one or more adverbs or prepositional phrases etc modifying the verb:
- It's raining continuously. / It's snowing really heavily out there.
- It's snowing for the first time in two years. / It's thundering on the coast of Maine.
But here, we have a noun group, cats and dogs, obviously used adverbially rather than as a plausible direct object.
- It's raining cats and dogs [out there].
With a typical extraposed-it sentence
- It's clear that the MP doesn't know the true facts.
a rearrangement fronting the semantic subject is possible (if unwieldy):
- That the MP doesn't know the true facts is clear.
As there is no semantic subject with weather-it sentences, this is impossible with say It's raining.
But at first glance,
_ Cats and dogs are being rained out there.
- Cats and dogs are being rained out there.
looks a possibility (after all,
_ Cats and dogs are being trained out there.
- Cats and dogs are being trained out there.
is fine).
But it's ungrammatical. The idiom (as idioms usually do) resists such transformations.
And that really includes, in standard English, deleting the verb.
The Farlex Partner Idioms Dictionary does admittedly give a parallel example (which may well have encouraged OP's example):
Wear a hat – it's brass monkeys out there tonight!
But it adds the caveats [usually British] slang.
So 'It's raining cats and dogs' is best regarded as an idiom best analysed at the idiom level, with 'cats and dogs' only in this usage a very strange adverbial looking like a noun phrase, and 'It's cats and dogs out there' a non-standard adaptation apparently using 'cats and dogs' adjectivally (so doubly resistant to analysis about 'plurality') but with 'Those are cats and dogs out there' not even vaguely acceptable in the intended sense.