How can I deduce/identify an app from a notification icon?
You need to investigate this on your own. There's a generic way to do this:
There are at least 4 possibilities:

If the phone has Android 4.1 or later you can long press the notification and see the App info:


Look that App icon up in the list of installed  apps on the phone
Settings-&gt;Apps-&gt;installed apps

Just click it and see what app starts

Go to the Play store and look through the installed apps on [phone] list
To access this you need to be logged in with your kid's Google account. On the phone itself this is automatically the case, on the PC you need to enter username/password.