I came across one boy when I was teaching music for a while in a secondary school, who not only had no sense of pitch, but also no sense of rhythm. His "Nellie the Elephant" was a thing of wonder! I have no idea whether he could eventually have been taught to sing, but suspect not. No more than you can teach colour blind people to recognise colours. There are IMHO some people who are unable to distinguish any difference in tones, and cannot tell whether the sound they produce matches what is required or not. Anyone had any experience with the standard tests - can't recall the name now - that asks you to tell whether two separate tones are the same or not? They all sounded exactly the same to this boy.

And perfect pitch does not mean being able to sing in tune. Perfect pitch means being able to pitch a note without outside stimulus - sing any named note on cue without relating to any aural input. I knew a guy who had that in a male voice choir I belonged to. We called him the human tuning fork.