Yes, but that will take a long time and much dedication, and I don't know anybody interested in trying to remember many notes. There is also the risk that the more notes you try to memorize, the less reliable it gets. It's a lot more practical to just remember a few anchor notes and put the effort into training relative pitch.
For the purpose of performing music (or any practical activity), I'd say a good relatively pitch and a good memory of a few anchor notes, is as useful as absolute pitch. Functionally, AP isn't much more valuable.
But I do think, based on observation, that there is something fundamentally different with AP. It's like they hear sounds very differently, and their brain has a special mapping of frequency that I don't have.
They recognize very subtle deviations in frequency. I have had a person tell me "It's too sharp" when I was only off by 3 cents according to the tuner.
They don't make mistakes and they are fast, almost naming the pitch as soon as they hear it. It's effortless, precise, and accurate.