R.
As a mother to an only child (who is 4!) I can say that this has nothing to do with being an only child. Correct me if I'm wrong, but aren't many kids "only children" at this age?? Maybe I'm reading too much into this, but I'm slightly offended that this has anything to do with the scenario.
It's common with kids who aren't big talkers to say "mine". They don't have the communication skills to get their point across and it seems to be the only way to do it.
I think this is a phase. I think mom is doing a fine job with stopping her and making her say "please" before she just grabs something. Praising her for sharing is also a good thing--which it seems mom is doing. This is fairly normal behavior. I've seen dozens of kids do it.