Don't overthink the Amazon Echo Connect. While the name is somewhat confusing given it lives alongside Amazon Alexa and Amazon Echo, the Connect really just another avenue for Alexa Calling. (OK, that's one more name you need to know, and thus adding to the confusion.)
Here's the real deal: Alexa Calling is the method by which an Amazon Echo device can make phone calls. It uses the internet to do so but uses your phone number and contacts as identifiers. (Though you can still have it show up as "Unknown" if you want.)
Echo Connect differs in that it ties into a landline instead. It's a separate piece of hardware (it runs $35) that you'll plug into an actual phone jack. And from there it'll dial out and connect you to whomever you want.
You can use the Echo Connect itself as a speaker phone, and it's completely separate from any other landline phones you have in your home. (That is, you don't plug the phone into the Connect .)
Or, once it's set up, you can use any other Echo device to play phone calls through the Connect.
Think of it like this, perhaps: It's a way to bring smartphone-level contacts to a landline.
Here's how Amazon puts things:
With Echo Connect and a compatible Echo device, you can ask Alexa to call anyone using your home phone service—all you have to do is ask. Because Echo Connect uses your home phone number, either landline or VoIP, friends and family recognize the call. Answer your home phone on your Echo from across the room, so you can easily talk to anyone hands-free when you're busy making dinner or away from the phone.
None of that changes the fact that it's still kind of a niche device. If you're steeped in this stuff anyway, there's a good chance you don't actually have a landline laying around anymore. But some folks do. And for that, there's Echo Connect.