How do I care for my wife and not lose my own identity? I am seen as her caregiver and nothing else.

You try and maintain your identity by taking time out to do things that you enjoy and by making yourself a priority as well as your wife. Yes both can be done. You may not get as much time away as you'd like, but any time away when you're a caregiver is priceless.
So make sure you're hiring some outside help to come in to give you a break, or have family members or volunteers from your church or elsewhere to give you breaks, and then get out and do the fun things you enjoy.
While I was caring for my bedridden husband, just being able to go to the grocery store and take my time was often just the break I needed. Otherwise meeting a friend for lunch or supper, taking a walk through my neighborhood or just sitting outside on my patio with a glass of wine was enough to rejuvenate my soul so that I was able to continue on this journey with my husband, and not lose me in the process.
The important thing to remember is that you matter too, and that you MUST take and make time for yourself. It is then and only then when you will be able to maintain your identity.

OP you might want to change your handle to think of yourself in a different way. Project how you want others to see you and maybe they will follow your lead.

You're stuck with that perception. During the counseling and vetting that accompanied my first family caregiving job, I heard the words from a social worker: “your past life (meaning my satisfying career) is over. You’re a caregiver now.” Yup. She actually said that to me.