Just my $0.02 on the topic as both an IOs and Android user (I now use Android devices only in the house and not as portable devices as I switched to iPhone/iPad for travel), I find this shortcoming of IOs embarrassing for Apple.
My wife and I have identical iPhones with each of us having identical bluetooth speakers and BT Keyboards. I actually have to walk outside the house if my wife speaker or keyboard is turned on and not paired. Either that or I have to find it, turn it off and pair mine. Since we purchased these devices when we were both on Android, it wasn't a problem. I named mine and she named hers. Easy peasy.
Since switching to iPhone, it is a royal pain. So much so that I purchased a new Keyboard for my iPad (I am writing a book), not exclusively for this reasons but it was a consideration (The new keyboard is a bit bigger and has a slot to keep the iPad straight, plus it is a 3 device unit, meaning I have a selector and I can use it to drive my iPad, iPhone and PC at the same time and switch among them as needed).
It's high time Apple addresses this. I can't imagine how frustrating this must be in an office where everyone uses the same equipment.