But, you have your backup that you used to migrate everything you needed to the new Mac
(Unless someone else set up your new Mac for you, like a tech at the shop where you purchased your iMac)
Here's something else to consider...
I thought that the bluetooth device name is stored in the system - however...
I don't use a bluetooth mouse very much, so I experimented a little, changing the name on a magic mouse.
The name was kept through a restart, so all looks good then. This was set up and renamed on a 2008 MacBook, running 10.8.5
I THEN took the mouse, shut it off, Moved to my main Mac mini, running High Sierra.
Connected to the mouse, and the mouse, on a completely different Mac, showed the same custom name that I had assigned to the mouse on a different Mac. I don't remember ever seeing that happen before. I just remember default names. The mouse does not appear to use default names. It uses whatever name that you name it. Even on different computers.
So, the mouse stores its name internally (on the mouse)
Mine is an older mouse (with removable batteries), so I assume if I left it for a period of time with batteries out, that the stored name would eventually disappear.
Could be nice to know how your bluetooth devices got renamed to System Administrator. I would be more OCD about THAT part, and is also why I suggest that at least a reload of macOS might be a good way to proceed (just to make you feel better
) (I would do a full wipe and reinstall, but if someone else did your initial setup, then that may leave you a bit out of the loop, and I can understand that, too.)