Well my headphones can pair with a couple of devices simultaneously, but that wouldn’t help unless you were wearing the headphones at the time. Input devices always have to direct to a single recipient, so it would only ever be output devices. Note also that Bluetooth can have quite a range on it. It isn’t something you can pare down the range on like an iBeacon, though because it is radio, many of the same sorts of things will interfere with the range.
I can think of some scenarios where the approach you have outlined might result in unwanted triggers even if you did have the ideal solution. As the lowest tech approach, could you put a note on your washing machine to remind you that you should first find where your phone is before you load stuff into the machine for washing.
A step on from that to help you locate it is that if you have an Apple Watch you can ping your phone from that and have it make a sound (I’d say loud’ish at best rather than loud though). If you have a more interactive device, you could use the Find My service to trigger a loud noise on your phone.