Battery replacement for US-region iPhone 13 in Egypt

It will depend on the service provider. They are free to deny service if they want to. They are not obligated to provide service on devices sold in a different region.

If you ask nicely they may be willing to replace the battery. But again, there is no obligation they do so.

Contact the services providers and ask them.

Sorry, but this is a user to user only forum, not Apple Support. Yes, regional differences often means service may be limited to the country of purchase. If you didn't live in the US, yet you bought your phone in the US, this is a risk you took. If you can't find any authorized service providers in Egypt who can replace the battery in your phone, just hop on a plane and come visit the US where you can get the battery replaced.

And next time, DON'T buy your iPhone in a country you don't live. Any cost savings you might have seen will quickly evaporate as you're now learning.

It may be possible. You can ask, but don't be surprised if you're told you have to take it back to the US for service.