I mean, he might be against taking a bite at the same place that someone else already took a bite from...like me. But he probably feels it will be offensive to tear that part off or dump it cause that'll mean he thinks she's dirty or disgusting.
Some people naturally don't like that sort of stuff...even if it's by your own lover. I mean, I'm like that. Even if it's a family member or person I like who took a sip from my bottle, I'd be against drinking from that not cause I'm embarrassed...but because it feels kinda gross.