Neither.

This is, first and foremost, a series that uses jungian concepts. In this particular case, it's important to have in mind the concept of Complex and Archetype, both of them explained by Edogawa in the extras of P4G and a little bit in Tatsuya's scenario.

An Archetype is the fundamental unit of the Collective Unconscious, the sum of all patterns and experiences that life has ever had and will ever have. Through it, they organized in psychis structures common but unconscious to every life form, but that ultimately determine how the mind works and can perceive things. They are the source of mythology and folklore as well, reason of why they have a mythological character (Aka They are the real thing).

A Complex is, to put it blandly, an aggregate of experiences anad memories that surround an archetype, shape it into a personal concept and view that allows the creation of the personal unconscious as well as the ego/consciousness. Personas (as well as Shadow Selves) are Complexes, the personal view of certain Archetype given by one's own memories and life, and they can be both personal or collective.

The personal complexes (Personas and Shadow Selves) are unable to be summoned by any Wild Card (or person that has access to the Velvet Room) becayse those are tied to the person itself. On the other hand, the collective complexes can be summoned thanks to being a reflection of a collective, not a personal thing; those are the cognitions you see in the palace as well as in Mementos.

The only unexplained case is Aigis with Orpheus, and the finding it on battle is likely just a mechanic in case you need it again since there's no compendium