This is something that has always baffled me. Do all people who sing off key know they are? If so, why do they persist? If they don't know, is it a hearing problem?
One of my brothers was in the field artillery in Korea, and his hearing was damaged in a way that he could not hear certain frequencies. Some peoples voice were all but lost to him, yet he could hear every word I spoke or sang.
I had 4 brothers and 3 sisters. All but one sister could sing, but she couldn't carry a note with a co signer.
A friend recently gave me a violin, need I say more? I hit a lot of sour notes but I know immediately and I redirect my fingers to find the right note before moving on to the next one.

Still baffled.