No, the old Tab Groups extension does something different. It allows you to switch between different groups like you can switch between virtual desktops.
You can't assign an already open tab as that would require cookie merging in a lot of cases, but there is a related solution. It took me a few weeks to realize what it was (but once I did, it all clicked, and works well now).
You select a container to open a new tab, then open what you want in it (type/paste link).
After you are on that URL in a container, the container control panel has an option to always open that domain (path too?) in that container. Check that.
Next time you paste a URL/link or follow a link to that domain, there will be an interim page asking you whether you want to open in the container the link came from, or the one you specified you always want it to open in. You can check a box to make that the default from then on.
Before I realized this is how it works, I was looking for a way to submit something like this as a feature request. After I realized it, the number of containers I used exploded, since I just default domains to always open in their containers and it's fairly transparent now.
As a real world example, I have a "Reddit" container, and a "Google - Personal" container (and a "Google - Work" for work G Suite stuff). I don't have Google default to anything (but I have some containerized pinned gmail tabs), but I do have youtube default to "Google - Personal". I have reddit default to "Reddit". I can view /r/videos and stream embedded and the views and tracking is in Reddit's container, but if I click to follow a youtube video to youtibe.com, it loads in the "Google - Personal" container and I can like or subscribe as I want.
At this point I have containers for banking, shopping, github+stackoverflow, facebook (I don't use it much at all, but at least this allows me to be logged in and worry less about tracking), Pandora, Reddit, Work/Home google, News (for a recent WaPo subscription), and I'm probably going to add a few more (including HN) within the next few weeks.
I really appreciate your taking the stime to spell this out in such detail. I deal with an insane number of open tabs (70-100 most of the time) and multiple discrete social media identities so this could be a huge productivity boost now that I understand it better. Thanks!
No problem! I actually spent some time trying to get the right type of access to submit a feature request for the containers addon for this issue (with problems exacerbated by teeing to use containers the hard way while using that access) until in double checking that I wasn't submitting a duplicate request I ran across the features I outlined previously on accident, so I know from experience the frustration of thinking is almost what you need and it would take so little extra work to finish it.
Luckily, the devs seem to have worked out a pretty good system, there's just a disconnect between what it can do and what's obvious on first usage, which is a problem that should only get better as time goes on.
The containers addon does grouping, sorting, color coding, auto group association, and sandboxed cookies. Is there something else you're looking for?