This is happening to me too since I updated to Mojave yesterday! I am getting new update notifications every hour or so and every time I run it, it is the same version. Flash is freaking out since I installed Mojave, and an app that I rely on is now not working due to Flash.
I branched your post to a new discussion topic as it appears to be different from the topic you posted to.
There are 2 Flash Player plugins for Mac: NPAPI plugin for Firefox/Safari and PPAPI plugin for Chromium-based browsers. Notification Updates will only update one of these at a time (which ever one made the check for an update). If you're receiving notifications every hour it would appear that one, or both, of these are installed, but not the most recent version. Please provide a screenshot of the System Preferences > Flash Player > Updates tab. This tab shows the installed Flash Player plugins and their corresponding versions.