The records do say "age" 21 and 23, though this may be in some longer years of another planet.
These are the biological ages of Dehner and Mitchell which - unless I missed something there - belong to the species of Homo Sapiens from Earth (if these figures were referring to anything else than solar years it would have to state so in the file because otherwise that age figure would serve no meaning at all and be totally redundant).
The series and films were made for human audiences measuring time in solar years, the same applies for the time elapsed between "Space Seed" and "The Wrath of Khan".
Theoretically Kirk's line could mean 15 years of time as he experienced it individually, but it's Khan remark that nails that down.
He was (and still considers himself to be) the "king" of Earth and I've no doubt, regardless how long it takes Ceti Alpha V to revolve around its star, that he was measuring his time of exile in solar hours, days and years.