Is the empty set (or any analogy) ever non-unique? I seems SO obvious that the empty set is unique, and yet every lecture on set theory and topology feels the need to state the proof. Why is that? 
Usually if something seems completely obvious, but the lecturer still presents the proof, it is a tell-tail sign, that one is studying the special case without knowing, and the property isn't true in the general case. 
 A: The question seems to ask about motivations of people which is not a mathematical question. However, consider a set theory which allows urelements (atoms), that is, objects which have no elements, yet are not the empty set. The standard proof of uniqueness of empty set does not apply to atoms. Also, consider the category of sets and functions. The empty set is an initial object in this category. Any initial object in any category is unique up to isomorphism. This is similar to the uniqueness of empty set in set theory.
A: This is not really a question about the empty set as much as it is a question about mathematical education, culture, pedagogy, and practices.
The point being is twofold:


*

*This is a very good example of showing a simple proof. We have some axioms, and we derive a consequence. This is a good way to exercise the students in how to approach a problem that seems obvious (which are usually very hard at first, because you don't know what you're supposed to do). This is a good exercise in understanding that proofs require you to understand the assumptions, their consequences, and what sort of routes one might take in order to verify these statements.

*There is no absolute notion of "obvious". And it is important that mathematics is built on strong foundations. It used to be obvious that all numbers are ratios of two repeat sums of the unit; but then $\sqrt2$ happened to be irrational. History is rife with similar examples, where people made mistakes that would be considered very silly nowadays. But at the time, due to the fact that "obvious observations" were left unchecked (partially because culturally mathematicians didn't bother with very precise definitions from the ground up). So again, this is a prime opportunity for instilling some good values into students and teaching them that everything requires proof in mathematics.
