I can't match my voice to the piano - the music teacher would play a note and I would try to sing it and fail. I certainly can't recognise any one note as A, so whoever asked that, it's definitely not something everyone can do! I can, however, hear other people be off-key, and I can hear the tune and sing it recognisably enough. That's not being tone-deaf, is it?
Is singing in tune something people can learn with practice? DH taught me not to pitch my voice sky-high whenever I'm singing and that's helped - he is a good singer with good pitch and can play songs on the piano from ear, although he can't read music. But I'm not pitching myself lower because I hear the note - I'm doing it because I've learned that my vocal range is relatively limited and if I start too high I run out of notes that I can sing. I can't tell if I'm singing the right note or not.
One more question - do you think a child who hears their parent singing out of tune will also sing out of tune? DD is like me. DS was singing the notes of Let It Go perfectly in tune when he was a year old (and sang Ded It Doh!).