rename blue tooth devices

There are two aspects to it, as I understand

1. You have simply assigned a Nick name for your bluetooth accessory using a Mac and not changed its name. Had you renamed a bluetooth accessory and paired it with another device then the new name must appear on all other devices. Presumably, I am sure that is not the case with the action that you have taken.
2. There are devices which offer renaming. To understanding it better; there is a set of information transmitted by a bluetooth device to identify itself: generally known as "SDP packet" which includes a universally unique identifier (UUID) and a list of other available services. What Mac does is, it keeps a link of the new name provided by the user to that of UUID (Usually the last part) received from the device. iOS device is capable of renaming such devices which offer services to rename / rewrite the last part of the UUID on the device itself.

Hope I was able to put my thought across through what I have leant through my experience.

I'm trying to do the same thing and all of the internet searches I've done show a name below the Disconnect and Forget this device options but mine does not. I'm not really concerned whether I truly rename the device or just give it a nickname I want to find it more easily. I was able to easily change the name of the device on my laptop bluetooth settings but in iOS13 there seems to be no option to change the name/nickname at all.