I completely agree that, in a singaround, songs should not be considered ‘owned’ by anyone, apart from their own compositions.
However, there is an element of courtesy, when a member has a very small repertoire. To take it to the extreme, if you knew that the only song a particular member could sing was, say, ‘Scarborough Fair’, and your turn came before theirs, you’d hardly be likely to sing it. More realistically, some singers, especially newcomers, do only know a handful of songs, and if 2 or 3 of those were sung by other members, who were aware of this, in the same evening, it would seem rather inconsiderate.