The seat of a faucet is the metal part inside the valve housing against which the valve moves, and it can cause leaks when it's corroded or bent. You can see the seat after removing the valve by looking inside the faucet housing with a flashlight, and if it's damaged, you may be able to replace it. If you can, you'll see a square or hexagonal hole in the middle that accepts a wrench so you can unscrew the seat. It's possible to strip the hole, but if you do, you can still get the seat out.
Look carefully at the seat with a flashlight before you decide that the wrench hole is stripped. If the hole is perfectly round, the seat isn't removable.
Try the seat wrench again, making sure that you're using it properly. It has a square bit on one end and a hexagonal bit on the other. The square bit won't work in a hexagonal hole, and vice versa.
Tap the wrench with a hammer to seat it deeper in the hole and try turning the valve seat counterclockwise. Spray lubricant around the outside of the seat and give it 10 minutes to penetrate before you turn the wrench again.
Remove the seat wrench if it doesn't work and find a flat-head screwdriver with a head that is slightly larger than the wrench. Tap the screwdriver into the hole in the valve seat with a hammer. This may take a few sharp blows.
Grip the screwdriver with locking pliers. Once the pliers are locked on, push down on the screwdriver with one hand while you turn the pliers counterclockwise with the other. Stop pressing on the screwdriver when the seat starts to turn and unscrew it with the pliers.