It's not the same meaning as in English, it means that you don't have any person to be with or to do something with. For example if you need to play some game like tennis :)
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.
This can be heard for example in some classes when you need to complete a task in pairs. Someone might not be able to find a partner and say the above sentence.
Since "I do not have a pair" is a sentence that could only be used in playing a card game like Poker and really misses the point of the Polish sentence, shouldn't "I do not have a partner" be an acceptable translation?