It really only needs to be on for a few seconds after pressing the button to connect to wifi, and submit the order. No need for it to be in sleep mode.
Even if it's Wifi, an ESP8266 only uses 78µA in deep sleep mode, which is how these buttons will spend 99.99% of the time, so it could last more an year on a single AA battery.
i think many people are missing this key point. this isn't a "i need this right this very millisecond" its more of a "i need this later today if its early enough or tomorrow would be fine" button. which a single press of a button would wake up the device, connect to what ever configured network connection it is assigned to, send some data to an API end point, go back to sleep. that could make the button last many years.
I thought so too, but they have a real sounding TOS and quotes from executives at the various partner companies that check out. Seems like a really involved joke from a company that is quite serious about a lot of ridiculous sounding things (drone delivery for one).
I can see it being a very clever PR April Fools stunt. I wouldn't launch a real product like this around April 1st. However, launching a plausible but fake one will grab a lot of attention, genius.
I think the battery situation could be fine. I have a wireless mouse that boasts 3 year battery life. Since a 'dash' would only ever need to switch on and send a quick signal when pushed (which for detergent might be 50 times a year?) I imagine it could live for years (say a few hundred pushes) on a single cell.
I know a gent working on e-ink tags for grocery store shelves. They have a similar energy usage, lots of sitting idle punctuated by doing something. He's gotten 5 years on a single coin cell. I can't imagine that this would be much different.
I mean, who will scatter these "buttons" all over their house, which will inevitably need new batteries at ever shorter and diverging intervals...