I'm not sure if you're looking for an explanation of why Sets behave this way, or for a simple solution to the problem it poses. Other answers dealt with the former, so here's a suggestion for the latter.
You can iterate over the Set's elements and test each one of them for equality using the equals() method. It's easy to implement and hardly error-prone. Obviously if you're not sure if the element is in the set or not, check with the contains() method beforehand.
This isn't efficient compared to, for example, HashSet's contains() method, which does "find" the stored element, but won't return it. If your sets may contain many elements it might even be a reason to use a "heavier" workaround like the map implementation you mentioned. However, if it's that important for you (and I do see the benefit of having this ability), it's probably worth it.