It sounds like you're still wondering why we all say "What?!" when you describe your hand-shaking.
Originally Posted by "HrtHsbnd"
Why is shaking her hand weird? She is treating this like a business, so why not shake her hand?
Your wife and apparently many of us have an intuitive sense that offering a handshake in that social situation is unusual. For expert guidance on when to shake hands, see: https://www.quora.com/When-should-you-shake-someones-hand
Originally Posted by "HrtHsbdn"
My wife was cold and distant with me the entire time... I was just trying to give her what she's giving me.. She wants to treat this like a business.. What should I have done?
No physical contact would've been better. I see your point--that she was acting cold and distant. Maybe the handshake was a passive-aggressive way of expressing, "It's like this is just business and not personal"? Only you know what you were thinking, but if so, it's clever but better is not acting out.
Originally Posted by "HrtHsbnd"
She absolutely refused and walked off. When I asked her why she wouldn’t do that, she said because it was weird. I don’t know why that’s weird.
Offering a single handshake was a faux pas. A second error was not just accepting "No". You say she walked off.. and then you asked? Did you follow, call out, bring it up later? When someone says "No" to a physical contact, they shouldn't have to justify it. We are masters of and control our own bodies.