Arutz is completely correct here. If you both take your last prize on the same turn, then it comes down to who fulfills a second win condition (in this case you fulfill the condition of him not having benched Pokemon).
If both of you (or neither of you) have benched Pokémon, then the game goes to sudden death, where you setup completely but you each only have one prize card. Whoever takes that prize card first gets the win.
Not sure what happens if you simultaneously KO during sudden death. There would drop be the same secondary win condition possibility, but past that, my guess would be that you play more sudden death rounds until someone wins, but I have no idea if that's accurate.