It can certainly be done, but again, the tricky part to grasp in this instance is that the Hive blockchain itself wouldn't be the one to do this stuff. Any interface or service could do this in a number of different ways depending on what type of contact info you chose to give them, using the API- which is a fancy way of saying "the piece of code that puts information into the chain for you and goes and reads information from the chain and brings it back to you". But the blockchain itself doesn't have "settings" for each user in the way that you describe, so it's down to a website or something on top to do the careful part of managing your personal contact information against ONLY the info you want a notification for. Discord simply tends to be a medium most people on the chain are familiar with, and it's a big piece of existing software that safely manages a way to directly contact you while protecting that information that's easy to work with.
It's sort of like saying the database that all of your country's financial institutions report into doesn't send you that text notification on a public message board that everyone can look at, just your own bank between you and them as a private text message. You had to give your phone number to them and they created a special service to relay that information to you while protecting your phone number- people around blockchain are very aware of privacy and security issues, and managing personal contact info definitely falls in there.