Aha! The searching for "bluetooth" got me to an "About Phone" page, on which I could set the "device name". Had my wife bring her phone over - ABSOLUTELY IDENTICAL Moto G7 - BOUGHT on the SAME DAY. Did the same thing with HER phone and on the "About Phone" page, NO PLACE TO SET THE DEVICE NAME... Same everything, build, dates, you name it.....
So, I tried doing the search for "device name" and hers found a page for System > About Phone (wasn't that where I JUST WAS?) on which we COULD set the device names.
SO, in future, when we're both in the car (as we were today) and both have our phones (as we did), we'll be able to tell WHO'S phone got connected...
Thanks for the help!