There is an extension that can make externally opened links to open in temporary containers.
Not quite what you want but could work, depending on what's your workflow and what you're trying to achieve.
There's also a software that acts like a browser and handles links, but instead of opening them, it shows you a list of browser's you have installed and you can choose which one to use. It also has an option to open said links in private windows (incognito) instead of normal windows.
Again, probably not the solution you wanted, but works for me.
Of you like any of the two solutions I listed above, let me know, I'll send you the links.
Okay, so long story short, I hate it when stuff opens in my browser, be it new browser tabs or browser windows. Those sites pollute my browser history and save some unwanted cookies etc. I'm very religious about my browser, and other than some chosen sites in normal tabs, everything else is in private windows.
The first solution to stop stuff from opening links in my browser randomly was this:
https://github.com/mortenn/BrowserPicker
It's an app that acts as http/https URL handler but does not open links, instead it presents a list of browser's you have and passes the link to open with any chosen browser. You can also choose to open in private mode. This makes sense because sometimes I want different kinds of links to open in different browsers. I wish this was built into the OS instead of forcing you to have a default browser.
This one is Windows only, but I know there exist some Linux alternatives, like Braus:
https://github.com/properlypurple/braus
Now having all links open in private windows might be great but there's one thing it doesn't solve. You might now about containers, the separate spaces for sites to open, it allows you to log in into a website twice simultaneously, in different containers, each will have their one cookies.
Unfortunately, private windows all store their data in a single storage, and do not support containers. You can't open two private windows and log in into two different accounts on a single website.
In case you need this, there is this extension that allows you to open links in your normal (non private) window, but they'll get opened in temporary containers automatically and those containers get destroyed with all their cookies and other stuff after you close those temporary container tabs:
https://addons.mozilla.org/en-US/firefox/addon/temporary-containers/
It has its advantages but also has drawbacks. Those sites will still be kept in your history. You can either delete them manually or enable the setting to delete from history automatically, but due to a limited browser API it does not work as one would expect. I can tell more about that if you wish.
Basically that's it. Temporary containers are great but not too polished yet, and there is a huge number of settings in that extension if that's your kind of thing. If not and you need some kind of "it just works" solution, I don't know what else to suggest you for now.
Feel free to PM me if you need help with configuring temporary containers extension.