Student or Learner
1) There is a rivalry between these two teams.
2) The rivalry between two team had started since a long time ago and is still ongoing.
3) The derby football matches refer to football matches between two rival teams from the same city or state.
4) Two teams from the same city always compete with and dislike each other and it is called the rivalry between two local teams.
* I just watched the champions league final, i am trying to improve my vocabulary and English with regard to football.
I think you meant to scratch out the word "since" as well. The rivalry started a long time ago...The rivalry between the two teams started since a long time ago and is still ongoing.