> I know for certain if you change the setting General > About > Name in an iPhone it changes what everyone sees when they look at their list of available Bluetooth devices.

> I assume other Bluetooth devices are the same, no?

No. The iPhone is allowing you to configure what name it broadcasts. But you cannot just tell another device what to broadcast. That device must have its own mechanism for changing its name.

For example, many Apple wireless peripherals can rename themselves after your user account once you connect them at least once. That has to be a function of the peripheral though, it's not performed by the device you connect it to (past telling the peripheral the new name, of course). Third-party peripherals usually do not have this functionality.