I frequently wear a pair of Bluetooth headphones paired to a computer. If I want to listen to something on my phone I have to re-pair the device to my phone.
Is it possible, through software or hardware, to have both my phone and computer connected in such a way that I can get audio output from either device to my headset simultaneously.
Yes, but you need “multipoint” headphones that are designed to do that. Low end headphones tend not to have that feature. Medium to high end often do, but make sure before you buy.