Make Object Invisible in One Scene and Visible in another scene upon click of mouse

I would like to know how to be able to click on an object in one scene and upon clicking on it, the object becomes invisible in that scene and visible in another scene.

This is completely speculative, but you can query the ID or name of the currently loaded level using the Application class, so if the object has been made persistent (i.e. using DontDestroyOnLoad()), you might be able to achieve this by setting a member variable indicating in which scene(s) it should be visible. You could probably also store the relevant information in a static member variable.

Beyond that, I think this would probably fall into the general category of 'game saving and loading', which has been discussed frequently both here and in the forums. (Note that using PlayerPrefs would be one option, especially if there's not that much data that needs to be saved.)

Perhaps someone else will have a better suggestion though.