What's the exact usage of "that of"

I think in the following sentence:

Adam's answer was similar to that of clergy.

"That of clergy" can be replaced by "clergy's" or "clergy's answer":

Adam's answer was similar to clergy's.

So, if my understanding is correct, can we say the following two sentences are grammatically equivalent too?

This is my book.
This book is of me.

Since in the first sentence, "of clergy" was indicating possession, I expect "of me" to mean "mine" and indicate another form of possession too.

That of is unnecessary when the comparison is to a single noun and you can use an apostrophe:

• Adam's problem is similar to John's.

You can probably also get away with it for a two-word noun phrase:

• Adam's problem is similar to his friend's.

But you need it for longer noun phrases. So, we don't usually say:

• Adam's problem is similar to the majority of people in the western world's.

but:

• Adam's problem is similar to that of the majority of people in the western world.

The following sentence is problematic:

• Adam's problem is similar to the majority of people in the western world.

because it is not the case that problem and the majority of people in the western world are similar entities.

Addendum:

That is a pronoun whose referent in this case is problem. Of course, if the referent is a plural noun, then that changes to those:

• Adam's problems are similar to those of the majority of people in the western world.

Clergy is normally preceded by the:

Adam's answer was similar to that of the clergy.

You can get rid of the if you add an adverb that indicates place or time:

Adam's answer was similar to that of clergy everywhere.

Yes, you can rephrase your sentence to be

Adam's answer was similar to the clergy's.

Saying "This book is of me" is grammatically correct, it does indicate possession just the same as saying "He is not of this time" meaning he doesn't belong to this time. However, it is a strange construction. It sounds stilted and possibly archaic.

I recommend against using it and would prefer "This book is mine".

Ditto Shoe on the direct answer to your question. But on a side note:

You would not normally say, "This is the book of me." You can say "This is my book", or "This book is mine" or sometimes "This is the book of mine." In English, the possessives of pronouns are special. We don't see "me's book", we say "my book". Likewise we don't say "the book of me" but "the book of mine".

You would almost always say "my book", but the "of mine" form is sometimes used for emphasis. It sounds pretty awkward when you're just identifying a possessive of an object. It's more commonly used when you are qualifying something with an additional noun. Like, "Bob is a friend of mine." "History 201, a class of mine, meets at 2:00." Etc. You could say "Bob is my friend", etc, and it would mean pretty much the same thing, though "is my friend" might mean that he is your only friend, while "is a friend" indicates there could be others. There might be other subtle differences in context.