Is there a way to give root privileges to an app through the ADB? The Android Debug Bridge (ADB) itself has root access (after executing the adb root command). Can that be used to give the same privileges to apps on the device? Is there a file somewhere in the system that keeps track of which user (app) has root access and can it be modified? If it helps I've already put the app in the system folder so it's regarded as a system app now.
Nope. Though you could use e.g. pm grant and grant it SUPERUSER, but that would have no effect if the app hadn't declared that in its manifest. And if it had, it wouldn't be necessary to grant it manually.

An app must actively request superuser privileges in order to gain it – and even then it only works with a superuser app in place (acting as "gate keeper") that then pops up a message for the user to confirm.