"Hey! That's MY song!"
The position that should be taken here is that no one "owns" a song. Songs are meant to be sung and karaoke is meant to be fun! We have heard of situations where someone has yelled "Hey! That's my song!"...and to make a point of your position, you could try an impromptu 'contest' where, over the course of the evening, 5 volunteers sing the same song and the crowd gives a fun and informal 'vote' at the end of the evening.
This being said, some singers have 'signature tunes' - songs that audiences have come to expect from them. Sometimes new performers will select one of these songs, which may bother the regular performer in the extreme. This situation becomes especially troublesome if the regular singer was scheduled to perform that song next. If you are in that situation as a regular, you could have an alternative song ready to replace the first one. Be courteous. You can always do the original song later.
In the same vein, it is never a good idea to deliberately perform someone else's signature song just for the sake of causing strife. Some singers, especially those with limited repertoires, can be hurt very badly by such poor sportsmanship.