As far as perfect pitch is concerned, I've got one opinion: Its not needed for any aspect of music and its questionable whether its even possible to learn or not (not enough evidence either way), its a complete waste of time to try and learn it when there are so many other things you could better spend that time on.
What is familiarity? The answer is interval familiarity or the other word, you need to internalize the interval. You are correct in relating the interval to a known song. As your as example said, My Bonnie, the first is sixth (Sol -- Mi). Therefore, the first thing to do is to remember all the interval. Once you master all the intervals (you can use your method by relating to certain song) then when you hear the same interval, you will be able to recall what you have in your brain.
On another point, the problem I've come at, having learnt intervals seperately from a music fragment and knowing it well, as soon as you try to figure out a song at the piano, you notice that a sixth may not be recognisable. Example: With "My bonny lies over the ocean" (there we go again ) the first interval is a sixth, but as a musical person you'll not recognize it because its not a sixth from the root but you hear it as a quarter till the root key, and an extra third above (or you won't hear it like that and be lost ).Oh got to go study for an exam....