Its ok to feel anxiety around something that seemingly challenges your identity, but you don't have to fight so hard to hold on to such an unimportant part of your identity.
Nobody is a programmer, really. We live in particular moment of the world where this new idea of professionalization tricks you into thinking sometimes what you are is first and foremost defined by what you trade on the labor market. And this is an especially potent spell for those, like us, who for right now particularly well compensated in this market. But either way you must try not to forget that this is just a kind of illusion. That we are more united by shared capacity than discriminated by "skill".
Why not take joy in the idea that what you have devoted yourself to, presumably, can be a more shared and understood thing for the world at large than that of a doctor or college professor?