Install the door hinges

This a bit of an enigma!

Most hinges have a removable pin that spans across the two sides. You stated you pin can't be removed did you try carefully hammering directly on the pin using a pin drift (a worn drill bit will do instead) that is equal or smaller than the pin you are hitting. You may want to measure the pin from both sides as it could be flattened or splayed out a bit on one side so you'll need to hit the other side to get it out.

There appears to be a small hole on the back side as you've noted in the second image could this be a setscrew? Using an Allen key can you loosen it? It maybe that simple! Once the setscrew is either loosened or fully removed it may then allow the pin to easily slide out!

It's also possible the mounting holes are threaded! As they used exposed screws on the door side. Then it's trying to figure out the screws diameter as well as the thread pitch used. Did the packaging give you any clues or gain that info from the supplier of the hinges.

You may need to create a template to test out the proper fitting of the hinge. Using a scrap of wood try to identify the first hole. then mount the hinge to then get the second hole identified. Using a piece of hardwood makes this a bit easier as you can the use the block as a drilling jig