Air Drop name of my new Mac not showing up correctly on older Mac running Catalina

I recently bought a new MacBook M4 laptop to replace my aged and slow MacBook Pro that is stuck running Catalina (10.15.7). I wanted to transfer files using Air Drop, which works, but the name appearing on the old Mac is not that name I set on the new. It defaults to the generic MacBook Pro. When using my iPhone 13 and new-ish iPad, the Air Drop name appears correctly. I've tried re-entering the name and restarting both Macs but that does not correct the problem. Is there something else I could try. Perhaps, the Catalina version of Air Drop has a bug that prevents the correct name from being displayed. One other factoid is that the Drop Box name appearing on all of the other devices is also MacBook Pre even though I've changed that name, as well.