Charger

Again, the iPhone 13 uses the same Lightning port as the iPhone 6. The fact that the cable works with your 6, but not with your new phone may indicate that either something is lodged into the 13's port or that it is defective. Regardless of the reason, I suggest that you visit the Genius Bar at your local Apple Store or Apple Authorized Service Provider to have them take a look.

Both the iPhone 6, and the newer iPhone 13, come with a Lightning port. You don't mention what exact wall charger that you got, and it would help to know as we can only go on what you provide us.

Based on what you did provide, my guess is that the "wall charger" is actually a power adapter and it may be one that has a USB-C port. If that is the case, you would need a USB-C to Lightning cable to be able to charge your iPhones.