Uninstall and reinstall any app you want their default provided notification sounds to be used instead of Samsung's choice of sounds. For example, the original Snapchat sound, or the casino sound when you receive an eBay bid.
This option was somehow lost in the SmartSwitch transfer to your new phone.
After reinstalling the app, the notification sound choices will read "APP PROVIDED SOUND", and you will still have the Samsung notification sounds to choose from.
Be advised, once you choose a Samsung notification, you permanently lose the option to change it back to the app provided sound. You will have to uninstall and reinstall the app once more.
I know this post is old, but I'm answering in the hopes that someone else who has the same issues sees this. I just restored to a Note 9 and had the same issue. Going into Settings>Apps, selecting the hamburger menu icon at the top right, and selecting "Reset app preferences" seems to have resolved the issue. I haven't tested all apps yet, but Messenger and Snapchat as the most obvious are resolved.
Worth noting that this does reset all permissions associated with each app as well, but it seemed like a heck of a better option than trying to uninstall and reinstall 169 applications.
It's a major bug/flaw/issue in Samsung Smart-switch allegedly. If you use SSS to copy from your old phone to a new one, you lose all the "App Specific Notification Sounds".
The solution... Uninstall and reinstall every single one of you affected apps (all 472 if you are like me, alebit not all of them have notifications etc).
It's been like this for some time now (at least since my S9, so several years), so Samsung want absolutely SHOOTING for doing nothing to fix this!
As I read this, I have just had to uninstall and reinstall SofaScore, in order to get my custom app provided "Goal" noise, for when a team scores, having recently just got a new Note20 Ultra 5G, and realised over the last few days, that yup, once again, I've lost all the custom sounds for all my apps.
Well done Samsung - utterly pathetic!