The default in Vector was set to be hidden by default and show them on hover (for UX reasons I believe). MediaWiki allows users to change this through preferences.
Either way, the behaviour is set and is expected to be reflected and not overridden.
I don't know if it is intentional (since it isn't consistently done), but the main link inside an Echo item in the overlay has CSS set to enforce the underline to be hidden at all times.
The "View edit" link on the contrary works as expected and shows an underline on-hover (e.g. it does whatever the skin or user has set).
The "All notifications" and "Preferences" links on the bottom also have this restriction set, though I think it makes sense for those as those are block level (the entire width/padding a click target), more like buttons.
For consistency the underline property should not be set to either none or underline for inline links.
This can be fixed by removing "text-decoration: none;" from the "#p-personal .mw-echo-title a:hover" rule in ext.echo.overlay.css (submitting a patch in a second).
I noticed that for the Modern skin it is going to be more tricky since that skin has a rule to hide the underline inside #p-personal (because the Modern skin displays the personal portlet links as buttons instead of links). Inside regular content (like the Echo overlay items) it is intended to behave like previously described for Vector. I don't see any way to reasonably fix this in Modern other than to move the Echo overlay container out of #p-personal so that it isn't an absolutely positioned block level element injected into a list item for the personal portlet links, but an absolutely positioned element in the same visual location, but outside of that list in the DOM (which is probably something that should be done anyway to fix other nasty edge cases with the layout).