Alex can be muted from the device itself. Go to the bottom right corner of the home screen, select the picture of the microphone and select "mic off" You can always go back to the microphone and turn it back on or manually activate Alexa if you only want to control that device without waking up others.
Other suggestion: if you have the ability to change the name of the Alexa service on your other device, that can be helpful to keep the controls seperated. I have Amazon, Ecobee and Bose Products. I have changed the name of the Amazon products to respond only to "Echo" while the rest will pick up "Alexa" it has helped tremedously as they are not all trying to respond at once.