The whole point of fingerprint is, like cookies, to track a unique identifier across multiple sites. It would be best to be unique everytime because that achieves the desired result of not having the same identifier in two separate places.
However I’m not too convinced this is really possible for every site because there are several ways of obtaining fingerprint of your browser, using different set of data points available some of which involve the very hardware of your computer. Things like your timezone, color depth and even the very fact that you are faking some values can be detected.
In short, theoretically a unique ID that doesn’t follow you around is most desirable but much harder to achieve than blend into the crowd. The best example of this being Tor Browser which you shouldn’t modify at all to achieve this effect.
In my personal and not at all expert opinion, it’s much better to compartmentalize your activities using separate browser configurations for separate purposes. Different addons, different privacy settings, even different window sizes. And if you can, even different devices. This way you generate different fingerprints that will be tracked, but never correlated to each other.
It takes some time to get used to this but actually is quite easy to do, and it helps me to stay organized and be more productive so for me it’s a win-win. I really hope I’m not getting this wrong cause I’ve gotten too used to this by now