If you are from China (going on the username) maybe the thought of being spied upon is a common anxiety, after all, your country does hold the world record for government internet survellance. So this paranoia from your boyfriend may have a social basis.
But technically, Safari has no ability to spy on people as far as I know. In my opinion, and it is just my opinion, I think your boyfriend is mistaken. If the cup is cracked, let it break.