It isn’t an identity. It’s just a thing a person did, for compensation, years ago.
A thing that only certain people do, and no one outside that small number of people actually realize what it was like and what life is like after. So having a “club” where you can go and hang out with people who actually understand what life was/is like after doing what you did, isn’t a bad thing.
I’ll repeat what another member said above:
The VFW is a club that hopes it withers away from lack of new members.