It is possible to develop this skill at a later age. We all have ears which are capable of it. For example we can recognise a note as very high or very low without any other note to compare them with, even if the timbre of the sound is identical. Recognising individual notes by name is just a more refined version of this same pitch recognition skill.
Many musicians find they have some degree of absolute pitch already. The A440 tuning note may be familiar, or middle C on a piano. Often guitar players find they recognise the sounds of open strings because they have spent so long hearing those notes while tuning their instrument.
To develop fully-fledged and reliable sense of absolute pitch takes quite a determined effort. You can read about how to go about it in these articles: