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.

[Renathras]

So I'll propose the following :
-Lily is not a living being, but the exact formula stored in crystal by the previous owner. maybe so she could persist as she had no material form

Note that I'm personally fine with Feo Ul being a form that SCH can summon, but I have an issue with the above.

In SCH lore, it's said that the Faeries were "bound" (the word used) to the SCH Job Stones. It doesn't saying their formula was discovered or inscribed, it says they were bound. Generally speaking, binding something requires that something to be pre/independently-existing. That is, there must be a thing to be bound.

This HIGHLY implies that Lily (her name) and all the other Faeries (like Lilac, the one from Setoto's father) were as well. It's more probable (since the SCH's didn't SEEM particularly evil) that this was less an involuntary capturing and more of a pact forming, that from then on connected the individual Faerie with an individual SCH Job Stone. The actual Faerie player SCH's summon is named Lily, with Eos and Selene being two sides of her personality (since she did have pretty bad amnesia when we found her), but she's one being. And for the record, Eos seems to be her preferred/actual form (it's the one she uses in all cutscenes she's in with us)

So a lore reasoning would more be if Feo Ul and Lily were to agree to merging their powers to help their WoL and agree to "share" a pact with you. (/Rand Al Thor intensifies)

It seems very unlikely that Lily ISN'T a living being (in some form), or that she's just a formula unless the lore we have is exposed at some point as being some complicated lie the SCH's of old established and perpetrated through centuries for reasons that...honestly wouldn't make any sense, as there's no "gain" for them by their enemies thinking the Faeries are formula spells rather than individual beings. Not to mention they were known to both Mhach and Amdapor, the latter of which we KNOW captured and interrogated SCHs (Setoto's father among them), and the former we can reasonably expect also did that. Meaning one or the other of them would likely have determined if it was a lie and exposed it, if it were, in fact, a lie.