I think "you can train someone to sing" is a bit like saying you can train someone to spell.
My dsis if you say a word, even one she hasn't ever heard or seen written down, she can spell it. Somehow she can just know how it is spelt. She is a natural speller.
I am not a natural speller. I can know a word well, seen it written many times and still struggle to get it close enough for the spellcheck to work out what word I wanted. However if I want to, I can learn a particular word to spell. It doesn't help me spell other words, but I can learn to spell a particular word.
Also:
My dsis is tone deaf. Give her a new tune she will drone it. If she takes the piece of music and listens to it over and over again, and tries to sing it over and over again, she can sing it just about in tune. But it doesn't sound natural, it sounds like she's learnt it note by note-perhaps like a small child who is reading by sounding out every word. Alter the tune slightly, and she has to relearn it totally.
I can sing in tune, I wouldn't say I was good, but play me a piece once to twice and I'll have most tunes safe to sing 6 months later. I can manage basic harmonies as long as the other person isn't singing too loud.
We had exactly the same musical background, also my dbo, who is also tone deaf. Dm sings nicely and was always singing, we had music records/radio on a lot.
Dh is interesting also on this. He has perfect pitch. He sings quite nicely now, he can pick up a written song and just sing it, play any note on the piano, he can tell you which it is, can tell you if something's off my a tiniest bit-in fact he finds that painful.
However he couldn't sing in tune until he was about 12yo, he said he could hear he was totally wrong, but couldn't get his voice to do it right.