The phrase "it's on" is used to say "goodbye" sometimes. In other words, it can be used as a response to someone who is leaving. It can also mean that something is about to happen which does not really match this context, I think.

you have changed the sentences, so much so that the question is now meaningless

"It is on" in this context means "I don't believe you. I think my team is better. And I think we should have a contest to see which one of our teams is really better at sports." It's kind of like the following: A: I am fast. B: No, I am faster than you. A: Prove you are faster than me by running to the end of the street. Let's start now. instead, person A could just say A: Oh, now it is on.

It is on = about to happen. Is the party on? It is indeed! Are we 'on' for tonight? We are!

I don’t think there is enough context to know exactly what it means. I think the second meaning mentioned by Diana makes more sense to me. When I first read this, I felt like they were arguing before this exchange. “A“ is trying to tell B to mind their own business and “B“ is saying “it’s on“ to indicate something is about to happen, an argument or physical altercation for example. In the US, one might say “it’s on“ after being challenged by someone else. Example: a: I bet you can’t finish that plate of food. B: oh, it’s on.