It would be more accurate to describe Buddhists as non-theists rather than atheists. Buddhists see no reason to believe in gods, whereas atheism equates to an absence of belief in any god. There is a subtle difference.
To the question. I don’t see how anyone could describe themselves as a Christian atheist or a Muslim atheist. Christianity not only requires belief in the God of Abraham, but in a messiah whose purpose is to bear the burden of the sins of humanity, thereby facilitating man’s only hope of salvation. In Islam an absence of belief in Allah as the one and only God is absolutely unforgiveable. Therefore no atheist could claim to be either Christian or Muslim.
Having said all that, if we’re talking about the alleged Christian philosophy of kindness, love, and forgiveness, I know many atheists who put their Christian neighbours to shame.