Dealing with my father's loss of identity, help!

Have you taken your father to see his doctor? Does he have a diagnosis of dementia? If yes, then his losing his memory of his identity is part of that disease. You can attempt to keep "reminding" him but this will be draining and unproductive. You can make him a scrapbook of his life (and the people he loves and who love him) that he can look through whenever he starts asking those questions. It will keep him occupied for a while but it will be Groundhog's Day shortly after that, maybe even the same hour of the same day.

I would also put my energies into creating activities and distractions for him that give him a sense of purpose: pairing nuts and bolts, sorting screws or washers, sorting and pairing socks, chopping veggies for dinner, polishing silverware, whatever he is willing to do that is safe for him to do with minimum supervision. For my 99-yr old aunt we have her fold a large pile of kitchen towels. We emphasize that it would "really be helpful to us" if she'd do it and she always does. She folds the same pile several times a day. We also have her sort colored poker chips (because we had them already) and she makes them into patterns. She was a graphic designer and artist before she retired. She remembers some of us sometimes and not always accurately, and not a lot about herself or her life.

May you gain peace in your heart during this journey with your father.