The problem is, the person saying Jesus is god in reply is not addressing the claim Allah is god directly. He's raising a separate point. I would agree both are positive claims, but that's not what we're really talking about in this thread.
If the Muslim says Allah is god and the Christian replies "I don't believe you", that is not a positive claim. Likewise, if the Christian says Jesus is god and the Muslim replies "I don't believe you", that is also not a positive claim. On that note, if an Atheist said "I know for certain that no god exists", that's also a positive claim. If either the Muslim or Christian challenges the Atheist, then the Atheist would have the burden of proof in that case.
He's trying to turn rejecting a claim into something that requires a burden of proof, when it simply isn't.