Welcome to Android Central! Which phone? What model year is the car, and are you using the factory-installed stereo or an aftermarket head unit?
On the phone, go to Settings>Bluetooth, tap the gear icon next to the car stereo, and make sure Contact Sharing is turned on. If it already is and the problem persists, then forget the phone on the car stereo, and forget the car stereo on the phone. Then go through the pairing process again, and this time pay close attention to the phone screen during the pairing process. If you see a window appear asking permission to share contacts, make sure to accept that.
If that still doesn't work, then open the Contacts app, tap Menu>Customize, and instead of selecting Show All, select each individual contact group you want to display. If you want all of them to display, then manually select every group, but not the "Show All" option. I recall having to do this with my old 2008 BMW 535i.