I usually refer to the people important to me by their name rather than by their relationship to me. Especially if I am talking to somebody who knows them or to whom I often speak about them.
If the relationship is important to the conversation then I will say 'my' - my sister, my partner, my mum, my brother, my friend, my nephew. I don't think it's particularly possessive - my mum has 2 other children (and finds more to bring into the family sometimes), my brother has another sister (and also often a girlfriend), my nephews have another aunt.
It is very rare for any of my friends or family to ask if a new person is a boyfriend. They don't ask about boyfriends because I spent the best part of a decade single and very happy and very vocal about how being single was part of what was making me happy.
The lack of labels did cause my SO a problem for the first 18 or so months of our relationship. I couldn't bear to be called somebody's partner or girlfriend so we referred to each other as friends - because we are. Regularly my SO would be told off by his friends who were upset that he was treating me badly when he called me a friend.
They assumed that he was doing this as a way of keeping the relationship casual and that I would much prefer to be called girlfriend or partner. He was regularly being told he was being horrible to me or being an idiot because he was treating me badly and I might stop hanging around with him. Eventually, I got over some of my problems with romantic relationships and now I can cope if he calls me his partner. I can call him my partner too if it's important to the conversation (still prefer to avoid saying it, though).