Some people are really weird about shaking hands with every man they meet, in any situation. It’s super weird, the first couple times it happened to me I thought it was just that person being weird, but it’s happened multiple times since.
While I personally will shake just about anyone’s hand if they make eye contact and seem confident in themselves, that doesn’t mean everyone does, so I’m kinda with you on this. Respect people’s preference. It’s just basic human decency.