If you can consistently sing a song in the original key, you have something very akin to absolute pitch. But maybe you have it at a physical level rather than an intellectual one. You have physical memory of what it feels like to sing those notes rather than aural memory.
This isn't uncommon. When I used to play keyboard for talent contests - now a dead art, EVERYONE brings backing tracks - I would often play a song introduction as printed only for the singer to set off in the (different) 'key of the record'. In fact, part of my craft was knowing what keys many popular numbers were recorded in so that I could immediately adjust when required!
Unfortunately, this type of absolute pitch generally came along with an un-tutored approach to music in which reading notation and even knowing note names had no part. It can be difficult to transfer the skill across to a more structured way of looking at music.
Absolute pitch comes in many forms. I'm a pretty well-trained musician. I don't consider myself as having absolute pitch. But play me A on a piano, I can usually tell if it's 'out'. To some degree, I 'know A'. But I wouldn't rely on it if tuning the piano (another skill that I've never developed).
I can't offer any definite answer to your question. Maybe you can 'learn' E, A, D etc. and so be able to tune a guitar without any reference. Maybe this will never be reliable. But what you CAN do is work on your Relative Pitch. 'This note is C, so sing me F♯?' A very useful musical skill, and completely learnable.
And, heck, who needs absolute pitch when it's so easy to load a tuning app onto your phone?