Unfortunately, no.
You will have to set everything up again on the new device.
But, that's an interesting idea. You can send the suggestion to Google using the "Send Feedback" option:
https://support.google.com/chromecast/answer/6292231
Google doesn't read every post in this forum but they do read and track all the feedback they receive that way. Just so you know, they won't respond directly to feedback so don't ask them any questions.
