Do I need a Korean passport or ID to verify my account when buying tickets?

No. There is no Korean passport for foreigners, and you do not need a Korean ID to buy. What the Korean sites run is 본인인증 (real-name verification), and that is tied to a Korean phone number registered in your real name, not a passport. If your account name, phone number, and card all match, that is enough. Your passport only matters at the venue, where staff check that its name matches the booking when you pick the tickets up.