***** NOT A TEACHER *****
1. "Peter does not have a degree like his friends." I believe that many native speakers would speak like that. Compare: She says that she's a native speaker, but she does not speak like one." ["one" = native speaker.]
a. In my opinion, you have written a grammatically correct sentence.
2. "Peter does not have a degree like his friends have."
a. In my opinion, many native speakers would accept your sentence and would even speak like that themselves.
b. BUT many teachers still insist that in formal writing, it is the rule to use as: "Peter does not have a degree as his friends have."
c. And, I believe, most Americans prefer to use "do" instead of repeating "have": "Peter does not have a degree as his friends do."
3. In fact, if you wanted to be very formal and elegant, you could also write:
"Peter does not have a degree, as do his friends." [the comma is SUPER important.]
Student or Learner