If we say:
It's a book of my mother's
How could this sentence be identified? Wrong, archaic, formal or something else?
The meaning there is entirely equivalent to "It's my mother's book". We don't usually use of-constructions for personal possessions in English, and when we do we usually use genitives for the possessor as well. So, while we might say
The Falkland Islands are an overseas territory of the United Kingdom.
We wouldn't say
I play computer games on the computer of my father.
Your father is a person; the United Kingdom is not.
However, when we want the possessed object to have a determiner, we can't use the genitive 's or a possessive pronoun like "my father's computer", because there's nowhere to put the determiner. That's because the possessive is being a determiner. Thus, it becomes hard to express the idea that it is just one of several of that thing that a person has. We can put one of in front of the possessive, of course:
I play computer games on one of my father's computers.
If there's nothing like that, it is ambiguous as to whether there is one or several:
I play computer games on my father's computer.
Does my father have more than one computer? I can't tell from that sentence.
If we need to be able to add more words, words that the possessive takes the place of, we can use the of construction. We can also use that if we want to change the way we emphasise different parts of the sentence. However, when we do so, the possessor is still identified with a possessive.
I play computer games on this old computer of my father's.
It was always an ambition of his to go to the moon.
However, if there's a plural possessor that isn't a pronoun, you don't need to make it possessive:
The greatest fear of disabled people is...
And sometimes where the possessor is an office or title, rather than a person in themselves, you can use of without a possessive even where you would use the possessive in the other construction:
Sentencing is the prerogative of the judge.
Sentencing is the judge's prerogative.
Oh, I imagine there are exceptions, but when you're talking about a person in themselves you generally use the possessive pronoun or genitive 's even when you are using of, and failing to do so will be seen as wrong.