Can I retain custom taxonomy data when uninstalling a third-party plugin that created it?

You could check the plugin files and see if the plugin author has registered an uninstall function with register_activation_hook or if there is a uninstall.php file in the plugin folder. I think the custom taxonomy and the terms should stay in your database, if the plugin doesn’t do any uninstalling. You can then register the custom taxonomy with your own function, using the same name, object type and args.

I think WordPress will give you an error, if you try to register a custom taxonomy with the same name that an active plugin uses to register a taxonomy.