To be really correct, you would say as "he." Nominative case is understood after "as." It really means "as he has" or "as he does" even when the verb is omitted. In spoken English we often do say "him." If someone uses "he" in this case, it should absolutely not be considered incorrect.
Guess I'm old school, but I think that , at the very least, we should get credit for the correct grammar. Maybe give credit for "than him" also, but it's a shame to mark the truly correct answer as incorrect. Some of us actually say "than I" in this case. Call it pedantic, if you will.
There are several ways to say it.
"Lei ha tanti libri quanto lui"
"Lei ha altrettanti libri di lui"
"Lei ha lo stesso numero di libri di lui"
"Lei e lui hanno altrettanti libri"
The "AS something AS" translates in "TANTO qualcosa QUANTO", but is less used than in English.
For example... As fast as possible -> il più veloce possibile
As soon as possible -> il prima possibile, appena posso/puoi/ecc...
I am as old as you -> Siamo coetanei/e
I hope I was clear. :) If you have some other specific cases in mind, please ask!
... sometimes only "Tanto" or solo "quanto"...
I am as tall as you -> Sono alto quanto te, Sono alto come te
I have as many friends as you -> Ho tanti amici quanto te
(Ok, tanto quanto, tanti quanto works with "as much as" and tanti quanti with "as many as")
Sorry for the mess! That's a difficult topic and I didn't check a grammar, maybe I should next time XD