If…..

[Calysto]

Let's study some Lore (correct me if I'm wrong on some of these, it's been a while)

Start with Arcanist's Carbuncle :
-Is an aetheric construct
-Derive from an ancient concept
-Can manifest it's own personality (Ruby, Tataru's)
-personality seems consistent for a caster ; the cause may be aether, dynamis or another "variable"

Now Summoner :
-Imbuing Carbuncle's magical formula with a primal aether create an Egi
-Comes from Allag ; does not include it's own summon (unless you count demi-baha)
-Phoenix doesn't makes a lot of sense lore-wise (we know it derive from an ancient concept, but it's appearance as a primal when/how it did probably involved an external interference (like an ancient bird-like fiery goddess ?) Relative to Summoner, at best you could go for "osmosis with bahamut" but that's a stretch

And Scholar :
-Allow you to summon Lily ; comes with it's personality and memories
-Comes from Nim where it was used in the same way we do. No information on a base concept for now.

Some additional lore that may be useful :
-Most familiars seems sentient with memories/personality
-The only familiars we know persists without their caster seems to be material ones (Cait, Gigi, Meteion)
-WoL already created one familiar : Anima's spirit
-Soul crystals can retain their owners skills and sometime memories

So I'll propose the following :
-Lily is not a living being, but the exact magical formula for a construct stored in crystal by the previous owner. maybe so she could persist as she had no material form
-WoL can temper the said formula (Eos/Selene form)
-It is possible one of those form is actually the "original" while the other use WoL "variables"
-Therefore, WoL could imbue the fairy formula with Titania's aether
-Since Feo can already travel to the source, It could be even easier for her if WoL had a vessel already imbued with her aether
-She could also alter the construct to leave it with bits of her memories/personality along Lily's

Unless I made a mistake, it would be consistent with the known lore and allow for more possibilities in the future.