To make it more formal, I am looking for striking historical examples of objects or concepts that were well known in a field and perceived as different, but later discovered to be the same. I am interested in how they happen and what impact they have, because they are perhaps the clearest symbols of scientific success, and the simplest cases of what Arthur Koestler called bisociation, "perceiving of a situation or idea in two self-consistent but habitually incompatible frames of reference", which underlies discovery process in general.
This is potentially a "big list" question, but I expect that striking "bidentifications" that had a major impact are very rare. Below are three examples of what I have in mind, but examples in any natural or social sciences would be interesting.
Hesperus is Phosphorus (Venus)
Perhaps the prototypical such example, the discovery is anecdotally attributed to Pythagoras (6th century BC). Careful observation of the night sky reveals that the bright star appearing in the evening after the sunset is the same as the one seen before the dawn. Had to be done before the motion of Venus could be modeled.
Brahistohrone is tautochrone (cycloid)
"But the reader will be greatly amazed, when I say that exactly this cycloid, or tautochrone of Huygens, is our required brachistochrone." With these words Johan Bernoulli in 1697 presented his solution to the problem that attracted the attention of Newton and Leibniz, among others, and launched the calculus of variations. Perhaps, in some part because Bernoulli's example encouraged the motivating idea that solutions might be otherwise known entities. His method of solution, the optico-mechanical analogy, was itself an example of bisociation, which played a role in discovering the variational principles of mechanics. The brahistochrone is the curve constrained to move along which a heavy particle will reach the end the fastest. And the tautochrone is the curve that the particle starting at any intermediate point of will reach the end of in equal times. Today we should be even more amazed that the fastest curve is the curve of equal times because even a slightest variation in the conditions of the problem destroys the correspondence.
Coal is diamond (carbon)
In 1796 after burning diamonds (!) in oxygen Tennant discovered that only carbon dioxide was produced as a result. Which meant that diamond is chemically identical to coal, a conclusion that evaded Lavoisier. A century later in 1893 Moissan stirred a major controversy claiming that he transmuted coal into diamond in his furnace. Today his claim appears more credible, and of course diamond synthesis is an industry.