The English “Don’t you have Robert's book?” was written as follows in French:
Tu n'as pas le livre d'Robert?
Why the “d’ ” here? Is it like the “ ’s“ in English?
The correct sentence would in fact be "Tu n'as pas le livre de Robert ?", and the "de" is indeed there to mark possession, as the "s". When the word after "de" starts with a vowel, "de" turns into "d'" to ease pronunciation, but it's normally not the case before consonants.
If you have indeed seen it written as you state, with the "d'", the intent is probably to emphasize that the person is speaking in a very informal and oral style (I would even say rustic). But it's somewhat surprising because enclosing the d sound between two r's makes it difficult to pronounce (the sentence as you wrote would probably be read "le livreu d'Robert" with the "e" in livre not elided).