Does he actually have powers?

it's hard to categorize, because to do so spoils the movie.

No, he doesn't have powers - he has a vivid imagination and lives in a fantasy world - he may even be schizophrenic

SpiltPersonality

It's almost a category to itself, but the broader genre to which it belongs is called "slipstream" by critics and serious readers of genre fiction.

There's a broad tradition of fantasy stories, and especially horror stories, where it's entirely up to the reader whether stuff was real or hallucinated. And of course you see this in films, too. Some classic Twilight Zone stories did this (although more often Serling would put in a twist where the main character found physical evidence that it was real). The recent film The Babadook is a tremendously good example of this: whichever scares you more, actual monsters or psycho moms, you can see the movie that way.

The term for this is "liminal fantasy."

Griff the Invisible is liminal science fiction. I can't think of another example off the top of my head, and for a good reason: it seems to be less satisfying to the reader or viewer. You need hallucinations or delusions that can be parsed in terms of sci-fi rather than fantasy or horror, and by the nature of sci-fi, we do want to know whether it's real. A big part of the appeal of vampires, zombies, werewolves, etc. is that they may or may not be real, but sci-fi gets its appeal strictly from its believability as real.

This movie would be easy to categorize as a straight drama if it weren't for the ending. Shared delusions are so common in psychology that they have their own diagnostic category, shared psychotic disorder, and a classic fancy term from older psych - folie à deux. All the stuff about Griff being invisible plays like classic shared psychosis. They are collaborating in creating the delusion.

However, as others have pointed out, if there actually was a parallel reality where this was real, they would appear to be delusional even though they weren't. And when Melody falls through the door, that is either hard evidence that there is a parallel reality where they have superpowers, or it's the all-time king of shared delusions, one they made up spontaneously together in a moment of joint emotional anguish. Did Melody actually open the door and fall backwards into his apartment, with both of them hallucinating that she fell through it? It's certainly possible, but it's a stretch. Of course, it's not any more of a stretch than the notion that they do have powers in another plane of existence.

And of course, it is immensely touching either way: either they are so in tune with one another that they hallucinated this together, or they are not delusional after all.

It's pretty perfectly liminal.

Now, the term "slipstream" has become a blanket term for works that cannot be comfortably classified as either straight drama, fantasy, sci-fi, or whatever. Slipstream proper is the stuff that actually leaves you feeling weirded out about how to interpret it: basically, you're reaction to the question, what genre is this, is "WTF?" So this is not "hard slipstream." We're not confused about what genre to assign it to; we see that it is poised perfectly between straight drama and sci-fi. But I have found the term "slipstream" to be very useful to encompass hard slipstream, the liminal, magic realism (where the fantastic element is culturally and socially normative), surrealism, postmodern fabulation, and borderline genre (tales where the fantastic element is small enough that one can argue whether it deserves the label).

Prepare your minds for a new scale of physical, scientific values, gentlemen.

She couldnt walk through walls. They are crazy. remember?

The reason she falled through the door to me I interpreted it that Griff started believing in his delusions again.

"If god ever existed then he's DEAD now"