Possibly a security measure? Schools are increasingly becoming targets for anti social elements of late. I wouldn't be surprised if the public, on its own volition, pressurizes the government to have stringent security checks in schools.
> Requiring a government ID to buy a prepaid SIM card.
You can't authenticate without Aadhaar OTP. And don't expect the government to send OTP to an unverified mobile number. That would put huge liability on the government if tomorrow you claim that the number never belonged to you. Biometric scanners are not ubiquitous, so the need for linking SIM cards for authentication.