We definitely are Azem, and the game doesn't even make it subtle.

The only question remaining is that are we the sundered reincarnation of Azem? Or are we directly Azem, in that at the end of our journey, when we are fully rejoined, do we travel back to Elpis and become Azem in that age.

I think the signs point to the latter and that we are in a time loop. Not only backed up by the fact that all the time travel in this game is time loop based, and there is lots of it, but also that Azem told Elidibus to go to Pandaemonium and meet us there. The only way Azem would know that is if they were us. Emet also says we look just like Azem, it's because we are one and the same.

Tenzen and Ramza were also previous reincarnations of Azem. When each they died, their soul was returned to the aetherial sea to be reborn. If we had ended up dying in the 8th Calamity, the same would have happened to us. I feel it's likely we will learn of another past reincarnation in the final part of the 24 man.