FWIW, I use an app called MacUpdater that can run and show a list of available updates, and then you can click to update them.. I've used it for a while now and it works great..
Anyway, for App Store apps, the Update option just takes you to the Mac App Store page for that app, and then in the App Store, it'll change from Open to Update.. That's been the only way I've been able to update my apps as they're not showing under updates.
Aaaaaaaand I just went to grab a link for you and it looks like they're shutting down support of their app! BOOO..
Could be like the same thing that happened to the Bartender app.
mac app to update applications
www.corecode.io