I found the video below really interesting. It’s very technical, but the idea is that these developers used a Raspberry Pi or similar device to interface with Alexa.
While the developer says that it is cheap Amazon Echo, you actually have to push a button to make it work. You also have to get a microphone for the audio in. And then there’s the speakers for the audio out. So by the time you are done, you really have something like the Amazon Tap, but without the battery.
And of course it is going to look like a mess of cables and circuit boards.
Nonetheless, this is a great proof of concept. If you are looking to learn more about various hardware and software pieces work together with Alexa, this would be a great place to start.