Indeed. We loved the song, at its time (hm... we should not say that, it shows our age...). Well.. Just to cut the lose ends, the Romanian equivalent of the "cats and dogs" expression is "plouă cu găleata" (textual: "with the bucket" - think about the already famous "bucket challenge", hehe).
The Romanian "plouă cu bărbați", if not for the song, would mean there are plenty of men, like imagine an interview or something where you expect also women (or only women) to come, but you see men only, instead. You could say "hm, plouă cu bărbați"...
Compare: "plouă cu tâmpiți" (it rains idiots) (or a textual translation of "the mother of all idiots is always pregnant"...) when you have that bad day... hehe
Many thanks (and a lingot), with your hint I found it in Wikipedia (https://en.wikipedia.org/wiki/It%27s_Raining_Men). No Romanian article at present, though. So the educational content is that while “to rain” can take a direct object, “a ploua” cannot and requires the preposition “cu” to indicate what it's raining.