It's possible if you configure your /home/ to use a separate partition. But you would have to do that in advance. Even in that case you'd have to reinstall it but the nice thing about it is you won't have to reconfigure anything. It will be able to get all your previous settings from your home dir.

You could backup your home dir before reinstalling then copy it over after for the same effect.

This is the most important piece of information. You should edit the post and/or title to make this more clear.

Well, that makes a huge difference to the meaning of the question.

I don't know, but maybe the login is held in a dotfile such as ~/.dropbox or maybe in ~/.config/dropbox or similar, and just backing up that (not to Dropbox!) would be enough to restore being logged in on a different system.

Ah ok. So its not so much the current files that you want to retain, but the ability to receive files locally through sync, when someone else elsewhere makes a change?

Sounds a bit like not wanting to remove the Netflix app because its logged in with the unknown password of an ex.

I would back up all the important files you need before you change anything.

You dont need the install preserved, you need the login session preserved. I doubt that it's even possible

1. Having a separate partition for /home might be sufficient since dropbox keeps the login details in ~/.config
2. Use a tool like rclone and run sync manually. Can backup the API key post the in-browser login. If you spend extra effort, you can create a systemd file to automate this as well