What is the exact meaning of deep in the following sentence:
Beauty is more than skin deep.
Is it an adverb in this case? If yes, what is the meaning? If not, what grammar rule is applied here?
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:
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."
deep is an adjective. skin is a noun which forms a compound with the adjective, modifying its meaning: it answers the question "how deep?"
In general, when a noun modifies an adjective, it indicates that the adjective inherits the properties of the noun.
As you can see, this grammatical construct is used not only literally, but also for metaphors and exaggerations. The likely reason is that it is very slick: only two words are used, but they carry a lot of meaning.
Note that this syntax is used for adjectives that denote something concrete and directly perceptible. For instance, "abstract" is an adjective, and "mathematics" is a noun which names a subject which is abstract. But we don't say "this is mathematics abstract" in place of "this is as abstract as mathematics".
Some of the combinations are canned phrases. For instance, we don't often hear "stone hard" as a substitute "rock hard" (but "diamond hard" and others are possible). However, there is a "stone cold" combination, which in turn is not substituted with "rock cold".
Yes, deep is used as an adverb in this context, and means "to or at a considerable or specified depth."
The sentence may mean that beauty is not something superficial (skin deep), implying that beauty is something deeper: a question of personality rather than appearance.