They are all arguably 'correct'.
1. He has the same hobby as I. = Grammatically correct, but awkward.
2. He has the same hobby as I have. = Unusual, but as Aisling says, probably an acceptable regional variant
3. He has the same hobby as I do. = Fine. The 'best' answer, in that it is the least controversial.
4. He has the same hobby as me. = Purists would argue that that object case pronoun ( 'me') is wrong, but it follows a pattern that is very widely used among native speakers. It's normal English - I don't have a problem with it.
If you're doing a written grammar test - choose number 3. In order of 'normalness', I'd order them 3, 4, 2, 1.