Q: And these songs are ones I have only discovered very recently. So my question: does being able to remember and sing a song in key count as, or at least approximate, absolute pitch?
A: to put it to the test you should try to get it to work while another song is playing, and compare it to obtain that song's key. but i will tell you something ironic. you will hear the experts say "no, that doesn't count, that's just sonic memory. the real absolute pitchers can do it instantly." But for me, here's the twist: every single person i've ever met who claims to have absolute pitch, i test them on it and they fail miserably.... by way more than a half step