I can't copy text from a Notification.

I noticed this when Install Updates reported a timeout giving the offending URL. I assumed I'd be able to simply highlight the URL in the Notification and then just paste it into Konqueror with MMB but it didn't work and neither did Ctrl-C

Is it just my setup or is this an across the board 'feature?'

I'm not sure if it is specific to the version of KDE Plasma you are using but on Kubuntu 18.10 I can select the text of a notification, right click, select copy and then paste elsewhere. I take it you aren't able to highlight the text? If you are able to highlight maybe try right clicking to copy.