Information Details Smartphone Wi-Fi or data connection Echo Loop uses your smartphone’s internet connection and the Alexa app for connectivity and other features. If not using a Wi-Fi connection on your smartphone, carrier charges may apply. Please consult your carrier for information on any fees and limitations that apply to your plan. Alexa voice control is supported on Android 8.0 and iOS 12 or greater. However, please note that certain smartphones are not compatible with Echo Loop for Alexa voice control.
Besides that just looking at the thing I wouldn't think it's reasonable. An active LTE connection would be too power consuming for the thing to last all day long. Relying on WiFi would require some system for automatically connecting to access points its never seen before, and somehow managing to get through landing pages that spell out terms and conditions (which would raise all sorts of questions about consenting to terms and conditions on the behalf of a user who isn't even aware they're even being presented).
Also, IIRC, Qualcomm hasn't come up with a new wearable SoC since 2016, and having had an Android watch with one of those SoCs in it, I can tell you it doesn't have great power economy. My guess is it doesn't even have a proper SoC in it, and it's just got the brains of a Bluetooth earpiece in it, coupled to the tiniest, cheapest processing core they could get their hands on to basically tell the Alexa app that it's not just another A2DP device.
This is probably why you have to push a button to talk to Alexa. Doing real-time speech recognition to listen for the hot word, or having a continuous stream to the app, would suck down way too much power to last all day.
Plus consider how Amazon describes its battery life: “about a day with intermittent use.” My guess is they struggled hard to keep battery life reasonable. One of the first ways to economize battery life and improve performance is to offload as much of the heavy thinking as possible to something else. Basically they're using the phone the same way phones use The Cloud™.
IIRC the Pebble platform did something similar. Apps ran directly on the watch, but all data was retrieved through the phone via the Pebble app. This allowed them to exclude WiFi and cellular modems from the watch, helping improve its already stellar battery life, and reducing the cost, which was one of its big distinguishing features.