So if you got the vaccination when you were 5, you’re okay with the school, even if there’s very little immunity left from it (why we get booster shots from other diseases - like tetanus, MMR, etc.) However, if you got the actual disease at age 5, and it’s in your medical records, you still have to go get a blood test to show that you have enough of the immunity left from it? Niiice.
And while some vaccines do need boosters, chickenpox doesn't need one. The vaccination normally gives lifelong immunity. That's why they don't normally give boosters for it.