I always wondered that myself. Even when I played in Capital 1, no referee ever checked our identity...
When I was playing junior football in Italy, and later on U21 futsal, we had a card with our name, team and association we were registered to, and that card was checked against our ID card before every match, but nothing of the sort here.
It might be different in Cap Premier or Central League, but it definitely doesn't happen from Cap 1 to Cap 14.