Neither is partisanship, paranoia, or tribalism.
Not the same, you don't have children. This is like thinking people will freak out over receiving an insurance card or a birth certificate. Sure, some may but people who are getting the shot will primarily be used to receiving shot records.
I’m not saying it would be rational. And I don’t know how old your kids are, but I only understand the analogy from when I was a kid. Based on my experience as a proud vaxxer parent I’m not sure they really do the cards anymore.
I have like a billion kids of all ages, my youngest is 20 months and yeah, we still get an updated record every shot, they are required for schools. I think whether or not they use actual cards vary by where the vaccine is given, Oklahoma County Health Department, still cards, our pediatrician in Polk County Iowa, it's a piece of paper.