I've also tried switching around all kinds of notification settings, reparing and resetting the snipping tool, but nothing seems to help.
It works fine when using the shortcut Shift + Win + S, but I usually start the tool by clicking its icon in my task bar because that's quicker when my hands aren't on the keyboard and I'm mostly working with my mouse. When using this method, there is no notification and the snipping tool hides in the background after taking a screenshot so I have to click on it again every time in order to edit or save the screenshot. Really annoying.