Not absolute pitch, but possibly an exceptional pitch memory, which can and will over time develop into absolute pitch.
You say you can identify the actual enharmonic key the composer wrote it in? An analogy would be a word that is spelled in two ways, one in American English, the other in British English, and your being able to identify which word they spelled it in in their mind while speaking it. It would seem to be impossible unless somehow enharmonic keys radiate a different characteristic color or aura, because composers are said to choose a key based on its inherent quality. If composers did distinguish and utilize specific enharmonic keys for their purposes, then that would be evidence of the phenomenon. Such an ability would be a rare "soul" perception to distinguish between the inherently true and false enharmonic, beyond what is normally radiated to the ears.
i don't see an aura but i do see certain keys in color but when i hear a piece i hear the intent of the key it lets me know what the composer is trying to convey in the piece so i can tell the difference between c sharp major and d flat major to me the intent is totally different same with a flat minor and sharp i can hear which is sharp and which is flat by the intent that i hear in the key
Tonal Synesthesia?