>I noticed that there's also a difference with the US sports and the European sport. Somewhat the opposite of math/maths. Any explanation for this?

I hadn't noticed that one. I think that we use both, depending on sense, or maybe not, I'll think about it:

"I went to the school sports day"
"She worked in a sports shop"
"He was always good at sport" / "He was always good at sports"
"He's a good sport"
"What is your favourite sport?"

Still not sure - do we say something different?

"Of all sports, I like tennis the best"