@dkirmis. Of course. This is how Joomla modules work. This is what I, and other users of my modules, do to use Easy File uploader for multiple purposes on one or more pages.
You must be doing something wrong, since I do this all the time. Maybe you can explain what you have done, and I can advise you.
Are you using the module in an article with {loadposition}? If so, are you specifying a different position that relates to the position in different modules?