You'd need to be able to generate a session for them on your other Joomla site. Joomla as far as I am aware has no RESTful API to do this from an external request. You'd have to develop a component that can do that then send a request to it from CB Auto Actions. It's significantly more trouble than it's worth to be honest as you're doubling the complication of session management.

I don't think Joomla has anything for that (CB doesn't for sure). CB Connect may eventually be able to do that in the future, but I've no idea if it's even possible at this point without rewrites to our login process.

To login on 2 sites at the same time you need the cookies for both to be generated and their session data inserted into both at the same time. I suppose you could manually setup a session row insert using a Query action, but I've no idea if that'll work. The cookie you could try generating with PHP, but I highly doubt any of this is going to work without actually using the API calls on the individual sites (thus needing a REST API to call).

In short I don't think it's doable without heavy custom coding. Better option would be to put both sites on the same database then use templates and menu items to try and have them look completely different. There could also be an extension on the JED that can help with this, but am not aware of any.

Btw, happy new year.

Happy New Year!