I can't really help you with multi-site scenarios, because we've never done it nor support it.

It's probably easier to have all your users on 1 website then simply use Joomla menu links to provide different experiences for the user. You'd simply use multiple domains pointing to the different Joomla menu items to ensure they land on the right section of the site. You can even assign entire Joomla templates to menu items so you'd have 3 templates each with their own module positions different from one another. Probably easier and more cost efficient then handling 3 sites with 3 databases and keeping them in sync.

Beyond the above I don't know what more to tell you as we've never done multi-site so I have no idea how any of this would behave. It's all simply in theory. The only multi-site I know that would work is my proposal above (using Joomla menu items to give 3 different experiences), but it's not really multi-site.

If you decide to sync databases then you'll need to move all of CB and CBSubs tables. You'll also need to move Joomlas _users table and the ACL tables. Even then I've no idea how this is going to behave.

Ideal approach is a Joomla authorization plugin that on login checks their credentials against all 3 sites to see if they exist on 1 of them. If they do then pull their data and register them using CB API if they don't exist, otherwise just sync the data and login. Can probably use profile update date to determine what sites data is most up to date.

Sounds Great! Unfortunately, this extension does not exist.

Correct, you would need to create it. All solutions I proposed except queries through CB Auto Actions will require custom development.

Is what you describe what you will try to do in the future with oAuth?

I don't know. At this time that's just an idea. We haven't gotten the entire design hammered out yet, but in theory probably.

If not, how much would it cost to develop this (or maybe even adapt Jfusion)? [you can consider this a pre-sale question if this goes beyond the scope of this forum. You can PM me if you wish].

We don't take development requests. You'll need to hire someone outside of our team to do this. You can post your project on Joomlancers or Freelancers like sites. Keep in mind this is nothing simple by a long shot and will probably cost you thousands. I don't know what jFusion is nor have I used it, but if it doesn't work on J3.0 and you're using J3.0 it can't help you.

The question is to know if and how this can be done with CB.

The simple solution to this question is try. Create some test sites then test it out and see if it works. Beyond that I've no idea as I've never done such a thing.

What you are saying is that, from a conceptual standpoint (since you have no experience of this), doing this with J3+CB would not be so simple, because all the CB and CBsubs tables and the Joomlas _users table and the ACL tables would probably need to be shared (or synchronized). Is this correct?

Correct, basically the only tables you wouldn't share would be content like Joomla articles, etc.. All core tables like Joomla users, all CB tables (can probably exclude some of the plugin tables if you wanted their content site specific), all CBSubs tables, and ACL would need to be shared.