Right away, Nick already knows that Silver is his rival? They only had one battle! No other contact whatsoever. It wouldn't be clear that they were rivals until later on, when they had their second battle.
"You don't have to be so...." Lyra begins to call out to Silver, before Nick's hand comes across her face, signaling her to stop. "Silver!" Nick yells, Silver turns around to face him from the edge of town, leading to Route 30, "next time we battle, I'll be the one who walks away in victory!" Nick yells to his newest rival, Silver smiles as he walks away.