Recently purchased an iPhone X here, but i didn't get a receipt as the seller told me that they will email it to me. Still waiting for the email for over a week now. I live in Europe btw so it's not an option for me to go back in person.

Your only option is to have a photo for your device that shows the serial number and to send it to the shop that you have bought the phone from

Apple always stand by their products, so you shouldn't have a problem, as previously stated the serial doesn't lie. Email apple and go from there.