The pre-installed email app will depend on the phone. From your first shot I guess it's a Galaxy S8, so this is the Samsung email app. And I guess you are using the Samsung launcher too (if you don't know what that means the answer is "yes" - the launcher is the app that draws the desktops and app drawers, and hence also the icon badges that you are missing).
There were other odd changes with the last s8 update, such as the Samsung message app losing the ability to set individual notifications. I've not met this problem before, but it might be something Samsung have done (since it's the Samsung mail app and, presumably, the Samsung launcher, this is a Samsung problem rather than an android one). I'd check the settings for both the mail app's notifications and these icon badges generally, maybe turn those off and on in case something didn't get set right during the update (which will have replaced the built in mail app with a new version but not third party apps, so maybe it just didn't carry a setting over).
Otherwise I don't have an S8 so can't compare with my phone, but there are plenty of S8 owners around so hopefully someone will be able to advise or test things more directly.
I might remove the second photo if I were you - posting your email addresses in a public place isn't something I'd recommend.
