To get a MANUALLY INSTALLED APK to show up in the Google Play Store in the UPDATES or INSTALLED section, do either of the following:
1.) Manually install the APK with the SAME APK filename that Google knows the app as (i.e. from the APK download site like APKMirror.com).
2.) If you CHANGED the APK name when you saved it, it will NOT show up in Google Play Store as an installed app.
However, IF there is an update to the app, you will have to do a ONE-TIME search for that app in the store and update it. It will give you the UPDATE option on the app's store page. Once updated via the store, it will show up in the store's Installed / Updates section from now on.
Why does this work? Because the UPDATE will be done thru the store and the APK name it's using during the update process is what Google knows the app as.
If there is NO update, do option #1.
I always add the App name and version info to the front of the filename so I know what it is for later use (because if I'm manually installing, I'll probably be saving it for multiple / future devices).
I also keep the original name at the end, so I can rename it back to the original name if needed. Android is the only O/S I've seen where the filename matters during the install process.