Does each LEGO set include a unique piece?
Most of sets doesn't contain unique elements, however there are exceptions. This (uniqueness) is especially common with minifigures or limited edition/availability sets.

The easiest set (polybag) to confirm the fact that only some sets have unique parts is to take 624210, which consists of 6 pcs. of iconic 2 x 4 bricks in Red, which were available in numerous sets. Another option, of a boxed set, is to take any set consisting of basic bricks, like 5576. Parts included in 5576 are widely available and can be found in multiple sets. 

However if you take something exclusive, like 41999. This set contains some unique parts available in this set only.