I am speaking here now in general about passport request, not just schools. As most Thais dont understand English having a look at your passport is the easiest way to learn or copy down your full name. As English speakers this is all solved in a couple of sentences 'whats your name?" 'oh ok what can I do for you?'. Easy! And as expats we all tend to carry our passports, copies of, or DL's etc with our name in English.
For the school, I find its a Thai thing. From way back when I applied for the job here from my home country, it seemed like every 5 minutes they wanted copies of my passport as well as loads of photos. Every time visa/work permit renewal came up it was more photos and passport copies but like 2 or 3 times as many officially required on the paperwork. It remains a puzzle where they all went but we used to joke that they were sold off on the dark web for sham online dating agencies in Nigeria lol!