The best replacement for rivalry is unnerve.

The best alternatives to rivalry are rated as following:
★★★★★  unnerve
★★★★★  opposition
★★★  bullying
★  intimidate
★  sibling abuse
★  mold breaker
★  poison point
  sibling bullying
  mold breaker better
  michigan

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Speaking about rivalry, one probably shouldn't forget about siblings.

Other search terms related to rivalry and their ratings are:
★★★★★ siblings
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Google Trends

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rivalry
unnerve
opposition