I often hear people referred to as "der Paul" or "die Barbara". So I wonder whether it is correct to say "Das Buch von Paul" or "Das Buch vom Paul"
Note there are two levels of colloquiality/regionalism in your examples:
The lighter colloquialism is the replacement of the proper genitive form
by a prepositional construct using von and the dative:
das Buch von Paul
which is somewhat accepted in standard German, but especially in dialects that do this all the time.
The second level of colloquialism is the usage of a definite article in front of a proper name. This is in no way standard German, but still common in a lot of dialects:
das Buch von dem Paul
das Buch vom Paul
The bandwidth of "how correct" both constructs/the combined constructs are considered to be varies from "colloquialism" over "regionalism" to "dialect", depending on what book you read.
Absolutely "correct" (by the book) is only
Allthough referring to "der Paul" might be common in some places, its not standard to do so. "Das Buch vom Paul" is something you might hear from some germans, but you should understand this as slang.