The Google Play Store is responsible for keeping Android Auto up-to-date. This is because Android Auto is packaged as a regular app rather than a core system component. The latter would mean only getting updates via major Android software updates.
In practical terms, this means that Android Auto should be updated automatically in the background like any other app on your phone. These updates can fail sometimes, though, or be delayed indefinitely if your phone doesn't detect a Wi-Fi network. By default, the Play Store won't use cellular data to automatically update apps in the background.
Luckily, you can easily check whether you're on the latest version of the app by heading into the Play Store and searching for Android Auto. This is where you'd see an option to manually update the app, as pictured above. If you simply see the Uninstall button here, it means you're already on the latest version available for your device. As a sanity check, click on the three-dot menu to the top-right and ensure that the Enable auto update option is selected.
If you're using Android Auto on an older phone, keep in mind that you need Android 9.0 or higher to receive the latest updates. That said, Android 9.0 was released all the way back in 2018 so it's not a problem for most devices.