I think Amazon is using very careful legal language here. They are not "recording" things, but they are analyzing things in real time to identify keywords.

For the device to recognize "Alexa", the microphone HAS to be on and listening for that word. So it can be listening and compiling a list of all the keywords that you say, but not actually keep a recording of things anywhere.

Devices have been doing this for many years now. They aren't technically "recording" because the audio is only analyzed for certain keywords, then deleted. You can prove this by playing a Spanish language video with your phone nearby. You'll start getting Spanish language ads because your device picked that up. It never "recorded" the Spanish audio, but it did analyze it and track in real-time.