Why does the devil go by the name John Milton?

You pretty much hit it on the head. Paradise Lost's author is indeed the inspiration for Pacino's character. But this is taken from the source material, the novel by Andrew Neiderman.

The allusion is fairly simple. It's thematic in that Milton, the character, holds that everything Man knows about Satan is a lie. That history has been told by the wrong person. He should know, he was there. By calling himself Milton, he claims to be telling the real story of Paradise Lost. The story we see is nothing more than a retelling of that as well. As Milton, he's attempting to change the story, by tempting the new Adam (man born of a god) into falling once again. He's trying to do what he failed to do the first time round.

Also he's a cocky bastard and doesn't think it matters that he took such a significant name. Look at everything he does, everything he surrounds himself with. He can barely contain the "truth" of who he is. His name is as obvious as his taste in home decor and music. He's grandiose and needs everyone to know it.

It's clear that the allusion is to the 17th century poet and author of Paradise Lost. However, you say:

But the poem hardly portrays the devil in a positive light and the real John Milton himself was a religious person and believed in Christian theology.

But, to the contrary — this is a misapprehension. The devil is widely seen as a sympathetic character in Paradise Lost, at least to some degree. It is possible that this is intentional on Milton's part, as an argument for sympathy towards human beings who also sometimes chose evil paths but can be redeemed. There's plenty of room to argue on this, as Milton's Satan is a complex and three-dimensional character. But, whether or not Milton actually meant that sympathy, the important thing is that many people recognize it. Romantic poet William Blake (a century later) wrote:

The reason Milton wrote in fetters when he wrote of Angels and God, and at liberty when of Devils and Hell, is because he was a true Poet and of the Devil's party without knowing it.

So, whether or not Milton had sympathy for the devil, it's easy to see why the devil would have sympathy for Milton.