That does look like Adobe's update notification.
Note that if you have Flash Player for different browsers (e.g. Flash Player ActiveX Control for IE, Flash Player NPAPI plugin for Firefox, or Flash Player PPAPI plugin for Chromium-based browsers) you'll need to update each one individually. This could explain why you're getting the update notification multiple times. Based on the screenshot provided, you're updating Flash Player NPAPI plugin for Firefox. The download page autodetects the OS and browser and offers the Flash Player for that browser. If you have other browsers, you'll need to repeat for those browsers.
To avoid this, opt into Background Updates (update option = Allow Adobe to install updates (recommended)). Background Updates updates all Flash Player types* installed on a system within 24 hours of a new version being released. Notification Updates (what you have opted into, can take up to 7 days to notify that an update is available.
* Flash Player types that are not embedded in the browser, which are updated by the respective browser vendor.