Legally they cannot, no. And sure, they could violate contracts and break data protection laws. But they’d be trading an enormous hit on their reputation, who knows how many hundreds of millions in educational contracts for, what, some basic demographic data on kids? Data they’ll almost certainly be able to get via other means once the kid becomes an adult?

I don’t like Google’s involvement in education. I don’t like Google much at all. But we should still be able to examine this stuff rationally without devolving into conspiracy theories that don’t make logical sense.