I haven't tried it, but it should...so long as you've previously paired the Echo with the phone in question. An important distinction to make here is that with Bluetooth, "pairing" and "connecting" are two different things. Pairing is the process of establishing a sort of relationship between the two devices that authorizes communication between them. This need only be done once (provided you don't later take steps to delete the pairing information the devices store about one another). After that what you do is periodically establish a "connection" between them, which is an active communications channel that is used to exchange information (like music data). This is important because the pairing action will require a Wi-Fi connection, whether you initiate pairing via the phone app or voice command. But once that is accomplished, subsequent connections may be established using only the two device's Bluetooth radios.
So, make sure your Echo and your phone are already paired before you leave on vacation and what you want to do should work.