Sure, she has the right to, but just because she can doesn't mean she should. She's obviously doing something
that doesn't respect her boyfriend's wishes or else he would have never made this thread. It has nothing to do with being controlling or possessive and everything to do with respecting her own boyfriend's wishes. He has the right to not be happy with her contacting people in his personal life, provided the only reason she would have in the first place is because she became midly aquainted with that person through him. Nobody is trying to say she is his property. The issue isn't her wanting to be friends with his friends; it's the way she's going about doing it.
It might seem like a bit of a stretch, but what if you came home and your bf was having a random, private phone conversation with your brother or your parents or your grandparents. Or what if he started phoning up your employer? Would you say "well he can talk to whoever he wants"?
Like you said, she might have thought the OP wanted the friend there, but judging by this thread he didn't. It's not a big deal. I would just ask her to please not do that again and be more courteous about me keeping these two world's seperate. Why would any of that annoy you? I find that a little strange. If I had a girlfriend I would probably not introduce her to my family or friends and would hope that she would take the hint. Again, it's no biggie. Everybody's different and some people just enjoy their privacy like that