The models listening for the initial hook of "okay google" or "hey cortana" etc are stored and executed locally on the device-- usually in a low power manner.

This comes with a set of constraints:

Certain activation phrases are more catchy than others, with hard consonant sounds and enough length for the alg to grab on to. Allowing users to pick any name opens this up to a high false positive rate, where the device would 'respond' to arbitrary inputs rather than specifically its name when called.

Training models on new trigger words and then deploying them back down to the device is in its infancy, and voice detection has a long way to go (another example of this is the difficulty around responding to a specific person's voice, with its signature sounds etc). I could see this happening in the future as the tech improves.

My understanding is that these devices have ASICs which listen for their names, the actual pattern for the name coded into hardware, meaning that you can't easily select a new name.

Voice recognition takes more CPU power than the devices have, especially if you are trying to prolong battery life, and I think the more complex commands are relayed to a server instead of parsed locally.

Siri isn't triggered by TV in the same way. Take a look at last week's keynote where Phil Schiller says "Hey Siri" on stage and takes a pause to note nobody's phone was activated by his voice.

Echo lets you choose from a small list of names. I've got one that responds to "computer" for instance.

They do. Echo, Amazon & Computer :p