My understanding is perfect pitch is something people are born with.
How does it manifest before musical training?
A person's experience and exposure to sound and music begins in the womb. It's reasonable to suggest then, that the more music heard, especially when the mother sings along, the more inclined the newborn will be to music and sound, and especially to the mother's voice.
As such, yes every person is born with a degree of experience and ability to recall sound, notes, music.
Learning, exercise, training, practice - still apply. Personally, as a guitarist of 15+ years, I can remember and hear notes in my head, and I can hear the C Major scale without any effort - fluently. I can recall the riffs and chords I've played over and over and hundreds of times.
We are born with the ability to recognise sounds, music, notes..., and we learn to recognise and identify sounds precisely according to the ideal (pitch perfect) and theory of music.
Naturally, every person is different and is born with different abilities and talents, motivations, and influenced by environmental factors. So, if you were born into a musical family, you might then become interested in music sound creation, and from an early age, you would develop the ability to recognise perfect pitch from an earlier age.
It is a practical skills (and an ideal) - and it can be demonstrated if, for example, a person can recognise changes in scale and notes instantaneously whilst improvising. It's reasonable to suggest if a baby was trained while in the womb, they may be born with very effective pitch recognition and vocalisation ability - and with formal learning thereafter, leading to skill, judged according to some objective assessment method and common (industry) standard.