it's safe to assume that Electrum doesn't get updated automatically. I don't remember whether or not there was an option to adjust update settings during the installation process. I don't have automatic updates turned on in Google Play Store because I handpick the apps that I want to update and when I want to do it. It might be different for people who have automatic updates on.
Electrum do not update automatically. Although, on Android, if it is downloaded from playstore (which is not recommended as there are fake apps on playstore as usual), apps are updated automatically on Android devices if enabled, also it can be set in a way Android can bring up a notification of available apps to download. But using updating it manually is the best and also verifying the PGP key to make sure the site is never compromised before the new update, this which is safer.