A Polka walks into a bar

I'm a Polish immigrant. I've had certain experiences in my life that no native-born American will have because of this, and even a few that no non-Polish-immigrant will have. In addition, I'm eligible to join certain organizations in some major cities that accept only Polish immigrants as members.

I also have some very woke friends, because I don't discriminate against people for having bad politics. :) It's not uncommon to sometimes hear things like "Only Native Americans are true Americans, everyone else immigrated here!" or something similar. This is mainly used in arguements against racists or anti-immigration bigots.

But, in theory, such a friend could say to me "We're all immigrants, only Native Americans are true Americans. You have no moral right to exclude me from your organization. Moreover, you have no right to claim I am not an immigrant! We are all immigrants! You are a racist bigot, and the world should know you deserve to be shunned."

Would a person who was born in a different country not have a valid complaint here? That there are in fact some material differences between the two groups, and that conflating the word "immigrant" to such a degree erases my experience, and the shared background I have with other immigants?

To be fair, I came here so young that this barely effects me, but my parents would be extremely put out, and I would be strongly on their side. Or should my parents be required to sit quietly for fear of being called anti-immigration bigots?