Learn English – meaning of “deepen relationship” and “deep”

While some of the definitions and idioms mentioned/linked are relatively close approximations, they all miss the specific idiom "in too deep", referenced here: (to) be in too deep

This generally references being involved in something so thoroughly that you become unable to extricate yourself (you can't escape); it is meant to allude to the experience of being in a mud bog or shallow quicksand (not enough to bury you, but too deep to escape), or to wander into a cave or woods so far that you no longer know how to return.

Note this is not precisely the same meaning as being "in deep water" or "being thrown into the deep end"; these latter two idioms indicate overwhelming challenge or great difficulty, or being forced to face a challenge beyond your experience or comfort.

I think we should consider skin and deep together as a single unit, skin-deep, and it works as an adjective.

To understand the structure, let's consider something simpler:

James is good.
James is a good man.

The two sentences basically say the same thing: James is a good man. Suppose that someone thinks that "good" is not enough to describe James. (James is probably a great person.) In other words, James is "more than good", they can say:

James is more than good.
James is more than a good man.

Obviously, good is an adjective describing James. This is the same structure as the structure of your example sentence: Beauty is more than skin-deep.

Here is how a dictionary defines the word skin-deep:

skin-deep
^adjective
Not deep or lasting; superficial:
'their left-wing attitudes were only skin-deep'

So, to understand the meaning, your sentence can be rephrased to "Beauty is more than superficial."