As for why John Wick and The Punisher would be fighting in the first place, it’s not difficult to come up with a plausible scenario.
Frank Castle has dedicated his life to killing criminals, and while John Wick retired from the assassin life years before the events of the films begin, he left quite a body trail during his formative years.
Even if John is now just fighting for his life and/or seeking vengeance against those who wrong him, his dog or his car, it’s not unreasonable to assume that Frank would still think John deserves to be eliminated for past crimes.