I'm not aware that in the 'tradition' there were ever dedicated 'music' or 'song' sessions?
As long as I've been involved in Irish music there has. But that's only since about 1971, I'm still learning.

I'e never heard of an Irish-only singing session - do they exist outside of Ireland? In Scotland or England, Irish songs will only be a part of the repertoire.

It's different with Americana, you would not be likely to hear any non-American song in that context.