In the sentence This is Michael's book. the word "Michael's" functions as an adjective (I suppose). My question is, what kind of adjective is it? Do we call it a proper adjective or a possessive adjective. From what I've lerned, a possessive adjective should be pronominal and I don't think Michael's is a pronoun.

In the underlined sentence above, did I put the comma correctly?
Should I put a question mark or a period at the end?
Philip is so right!

I would not call this an adjective at all, but a noun in the possessive case (i.e. showing an ownership relationship to another noun)

A proper adjective is an adjective that is capitalized:

I like Japanese woodcuts.
The Hispanic culture is strong in America.
This rock is from the Pennsylvanian era.
Many stars with Jovian planets have been discovered.

A noun modifying another noun is a noun adjunct.
I like ham sandwiches.

There are some references to possessive adjectives, but these are applied to pronouns to distinguish the possessive pronoun forms: e.g. my (possessive adjective), mine (possessive pronoun).
You will most likely get a wide variety of answers on this. My simple way of looking at it is as a possessive adjective.

As for the punctuation, you'll get various answers, as well. In formal writing, it might well be My question is: what kind of adjective is it?
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 AlpheccaStars's reply was promoted to an answer.
The answer was simpler than I expected. You're right, it's a possessive noun. Why did I forget that?! It's high school English where I live.Emotion: geeked And possessive nouns do act as adjectives; hence the apparent confusion.

Michael is a proper noun, so it is a noun in its possessive case.
Michael's shows the ownership of a person (noun) so it can never be a possessive adjective.
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Some grammar texts describe possessive proper nouns as a determiner.

The book (The is a determiner)

Mike's book. (Mike's is a determiner.)