Ooookkk. Samsung phone, in the messaging app touch the picture icon and choose (i), touch notifications, touch sound. On my phone it only shows Zedge, so you are limited to Zedge tones. Search for whatever you need, notification, find the tone you want, touch the download icon, choose set notification or media folder. I don't know where it goes to a media folder at this point, but it will show up in the Zedge list of tones you have chosen.
I also found this to be the case, so I used Zedge -my only option, not my list of notification sounds- and downloaded a bunch of sounds, and then reset my default. HOWEVER, I still can't attach sounds to the majority of my messaging contacts. I don't even have the option. Also, I still can't get it to read my messages out loud.
Somehow, it is tied to whether you have SMS turned on with encryption. But today that doesn't seem to make a difference. I cannot change an individual in messaging. Google messages absolutely sucks. I used to be able to get it to read me my messages and have my internal sounds attached to people so that while my hands were busy I would be able to know who was texting me and what they wanted. Not anymore. What a piece of....