Edit: I assumed you meant that you were defining it for English. I think I understand now, you defined it for Polish! Thank you for that!
Original comment: I disagree. My first thought was that this is about socks. "I do not have a pair of socks." You could also say, "I don't have a pair of underwear." If someone asks to borrow your socks, you will probably just say "I do not have a pair."
I do not have a pair doesn't have anything to do with you being a part of any pair. If you have a mate, then you wouldn't say "I have a pair," you would say, "I have a partner/girlfriend/boyfriend/mate/etc."
Oh! And it can mean something else, which GuyRogers refers to.