Have you verified that your car's internal storage can handle music saved AAC? Many cars need MP3. If yours is one of them, you can convert owned music to MP3 in Music (app). Check your manual to see the proper format.
If your car can handle AAC, I'd suggest a simple experiment. Locate a song or two that you may have ripped from a CD at some point in the past. That will definitely be removed from Apple DRM anything. Copy that to whatever you need to then move it into the cars storage. Usually that is done on a SD card but maybe your car can work with a USB drive. Whichever the case, you need to be sure the card or USB drive is formatted in a format your car can read (which is different than the song file format itself), save the music onto it, then import it into your cars storage.
If that works, repeat the experiment, but this time choose a few songs purchased from the iTunes/Apple Music store. See if they will import and play from your car's storage. If so, just move all the music you want to move. If that doesn't work, you'll know that you can't directly copy the iTunes/Music purchases and have them play. So then move on to other options- conversion for example- to get them into your car's storage.
My own car came with a USB port in the console box, so I dug up an ancient iPod classic, synched a bunch of favorite playlists to it with Music (app) and now it functions as the primary music jukebox for my car. That works just fine. If you have such an option available to you, that might be another way to go using an old iPod or iPhone.
