#1 Hero doesn't like how the boss runs things
This is possibly the biggest similarity between these two rivalries.
In Austin's case, he was a rebel who didn't like how Vince McMahon was running the company. The fact that Vince wanted Austin to act in a certain corporate manner didn't seem to thrill the anti-hero inside him, leading to a Stunner that kicked off a legendary rivalry. Austin clearly didn't like how Vince was running things and wanted to do everything his own way.
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Then, we have Owens vs Shane, with Owens getting sick of how Shane is taking advantage of his authority, and putting himself in a position of power when it comes to on-screen storylines. Shane recently beat the likes of Roman Reigns and The Miz in singles matches. Owens' recent promo showed how he is done with Shane running things backstage so as to highlight himself on TV, and wants to put an end to it.