We have similar rivalry games like this between colleges and things like that. Most of those are within the same state though, not within the same city. I think there are some more regional rivalries though on a High School level for sure. Actually, the city I grew up in has two large high schools and they are probably each others biggest rival even though they are in the same city and part of the same school district.
High Schools don't kill each other though so there is a little difference. Having a rivalry with another school is good as long as you consistently beat them.
Ah, but you forget this is the US. They do sometimes kill each other over much less...