Shouldn’t I receive the old auto part after replacing it

Most garages will recycle old parts, but you said you asked for your old starter back. Starters are components that can be rebuilt , like alternators. So there is usually a core charge when purchasing a new one, so the part supplier gets the old one back. I assume you were charged this core charge that the garage didn't receive. Why did you want the old part? Are you going to rebuild yourself?

TDolby is right plus it's not common practice to return the old parts to the customer unless they request it. My Father used to do that to make sure he was getting the new parts he was being charged for. I think if you don't trust the repair facility you should probably go somewhere else! Hope that helps! Jim