There's more songs around than singers. Plenty enough for all to have a few of their "own" if they want to. I would personally think it the height of bad manners to sing a song at a session if I knew someone else was in the habit of doing it. Likewise it would be the height of bad manners to comment if someone sang "my" song, whether they knew or not.Mind you, if they sang it badly, and I'd had a couple of pints I might think differently.
These things sort themselves out . In pubs Ive frequented where oldtimers sing regularly I would say the conceptof song "ownership" was very strongly established, and didnt need to be spoken about.It was respected.