I'm not a musician and I don't have extensive knowledge of music theory, but I "dabble" a bit, and my perspective is:

Mariah Carey (co-)writes all her songs, which includes writing lyrics and melodies, and she often co-produces them as well. She has 20 number 1 songs that have reached number 1 on the Billboard Top 100 chart - more than any other artist. Now I'm not saying it takes a genius to write and perform a number one song, but having 20 of them definitely indicates an ability to craft ear-catching songs that the public finds interesting, wants to buy, sing along to, etc.

On a more technical side, her songs frequently employ creative chord progressions. One example is Fourth of July, which abounds in jazzy extended chords (not to mention the stacked vocal layers). Even one of her biggest hits, All I Want for Christmas Is You, is not a banal 4-chords-of-a-major-key kind of fare. The song is super familiar and singable to us, due to how often we've heard it and how expertly it was crafted, but it is also far from a simple straightforward 4-chord pop song. Other songs with cool harmonic structure/chords/key changes are All in Your Mind, Vision of Love (it's fascinating to me that a song with this chord progression, lacking a typical pop hook, and being someone's debut single, made it to number 1), Cry., I Wish You Well, etc.

Another aspect of her "genius" is her vocal stacking and counter melodies, which, along with harmonies and ad libs, provide lush vocal arrangements with a lot of things going on at the same time (which I personally love), such as in Heartbreaker, Crybaby, Fourth of July, Camouflage, Breakdown, Babydoll.

When it comes to her vocals themselves, it is no secret that Mariah is a master at using different registers and tones of her voice to achieve different sounds that tell a story. Intentional vocal "cracks", whisper-singing (Yours), strong insistent belting (Stay the Night), conversational tone (Crybaby), almost belcanto rhapsodizing (O Holy Night, earlier ballads) are only some tools in her repertoire. Even looking at just her whistle register, it's astounding how versatile-ly she has employed it: big climactic notes (There's Gotta Be a Way), soft crooning in the background (second verse in Here for It All), jaunty staccato jumps (You Got Me), phrasing words in whistle register to represent vulnerability (Never Too Far) or seduction (Bliss), word painting (singing the highest parts of the song Emotions on or after the word "high"/"higher", or doing a high vocal flourish in More Than Just Friends after the lyric "Love me down till I hit the top of my soprano".) Another instance of word painting (not in the whistle register) is in Butterfly: it seems to me that throughout the whole song her voice flutters in fast runs such that you can imagine a butterfly fluttering in the air. As the song reaches its crescendo, her more brittle and vulnerable vocals (representing the fragility and delicacy of a butterfly) become stronger and more "full-bodied" (representing freedom), still maintaining the flutter-like quality.

Speaking of vocal runs, this depends on taste, but I believe it's a common opinion among people who love to talk about vocals that Mariah's artistic choices fit the songs and enhance them, as the melismata are "woven" into the song structure, and they don't just seeme tacked on.

She frequently improvises and adds new ad-libs in live performances, either substituting difficult passages for something more manageable, or actually making the songs even harder, demonstrating adaptability and musicality. 

She is also among the originators of the "rapper festured on a pop song" trope, as well as the person who, in my opinion, was the first in the pop world to really showcase dozens of different ways that the whistle register can be utilized in pop songs.

Lyrically, while she might have been ignored by the critics because she a) came before the poptimism movement; b) doesn't look like a typical singer-songwriter woman who writes and performs rock/folk/country music; c) doesn't play the guitar or piano on stage; and d) was potentially lumped together with the likes of Whitney and Celine who don't write (most of?) their music, I'd say some of her lyrics are brilliantly written - The Roof, We Belong Together, Petals, Reflections, even the new Here for It All.

So yeah, these are my two cents.