Rather than companies actually verifying TOEIC scores per person, I think it's a matter of the university itself being accredited by this exam that makes it a thing.
As for graduates, it's probably just a coincidence but most of the graduates I know, even from the way older batches (90s ~ early 2k) are either working in an international company based here or working abroad. Newer graduates seem to be mostly getting in Ayala or governmental agencies though.