The entire first season is an examination of identity diffusion and foreclosure set against (or perhaps in response to) the classic nature v. nurture dilemma. And if you want to get really meta about it, one could argue that, if the Kelvin films are anything they are
strictly a classic nature v. nurture dilemma. But there are also elements of body integrity identity disorder. Both of which are struggles Fuller - and other prominent members of the staff - probably battled in their lives.
In terms of allegory The Bible (or rather Dante and Milton's interpretations of the Bible) factor much more prominently than Alice's Adventures in Wonderland. But I think the story is meant to reflect religion as a whole, as Micheal plays a prominent role across all three "major" religions. It should also be noted that all three have always depicted Micheal as a white male, but Fuller specifically cast a black female, which directly relates to the themes listed above.
I think a good simple example from Paradise Lost's Wiki page is
After Adam and Eve disobey God by eating from the Tree of Knowledge, God sends the angel Michael to visit them in the garden. Before he escorts them out of Paradise, Michael shows them visions of the future...
Insert Stamets and Tilly for Adam and Eve as needed.
There are similar examples relating to Micheal's story in Inferno to the MU arc, but I can't be bothered to look them up. But suffice it to say Micheal slays Satan, who if I remember correctly (It's been since Junior High.) had been presenting a false face to him. In modern storytelling contrivance, they just used a red herring: That is the arc angel Gabrial, Micheal's trusted ally.
It could be argued that identity foreclosure is the leading cause of the prescribed social dysfunction - or Peter Pan syndrome - that's become prevalent in the younger generations. People are starting to realize at a much younger age that the old social pigeon holes are total bullshit. Yet they can't reconcile this newfound revelation because of societal pressures to function and survive, Micheal's journey is an exploration of this. She's a human who thinks she's a Vulcan trying to fit in with Starfleet and endlessly struggles until she thrust into a world that is both a figurative and literal reflection of all that she thought was true and realizes the only thing she needs to be is herself. Voq-ler experiences a similar journey, albeit a much more literal - and infernal - one. (And here again, I should point out they deliberately made him an albino Klingon.)
There is a more cynical take here as well: that is the whole journey is meant as a reproach against pray the gay away.
As another note, Leviticus is littered with references to hooven grazing animals or cattle as being the barometer (Can ganglia also predict the weather?) of what is "clean" and "pure." Leviticus has, of course, more recently become infamous for its other proclamations of cleanliness and purity.
I think all this religious allegory is important considering that religion represents the largest conceived purveyor that is given charge to those social archetypes. They invoke a prominent religious figure, send her through storylines echoing her counterpart all to ultimately assert all of the "lessons" those old tales are meant to teach are wrong -- or at least should be taken as absolute.
And, as one more point of note I'll mention that Alice similar experiences an identity crisis that she ultimately remedies through the manipulation of numbers (or playing cards) like any good Vulcan might. Yet Carroll, despite being an advocate for reasoning and logic and mathematics was, at the end of the day, a man of the cloth.
Sometimes people just don't like something.
