There is no way to do it on most Bluetooth devices. The reason the TV can is it's routing only ONE of its audio channels through Bluetooth.
There is one possible way you could accomplish something similar. If you use the audio out port with a splitter. One plugged into a speaker at the Echo, the other into a Bluetooth transmitter which you could then connect to your bluetooth speaker. Technically the sound won't be coming directly from the Echo, but if the object is to get the sound in two different places, this would work.