Britannica is terse:"Möbius discovered this surface in 1858. The German mathematician Johann Benedict Listing had discovered it a few months earlier, but he did not publish his discovery until 1861". Möbius never published the discovery, it was found in his 1865 memoir and published posthumously after 1868. Derbyshire comments "It had previously been described by another mathematician, Johann Listing, also in 1858. Listing published, and Möbius didn’t, so according to the academic rules it should really be called “the Listing strip”. There is no justice in this world".
I am not sure how Britannica's "few months" are known, considering that publications came years after, but it is not unheard of for two people to make the same discovery almost simultaneously, when an idea is "in the air". Newton-Leibniz and Leverrier-Adams are famous examples. What seems to be missing is what exactly was in the air, how did the Möbius strip come up? MacTutor says only "this discovery was made as Möbius worked on a question on the geometric theory of polyhedra posed by the Académie". This is more or less repeated everywhere else. But the context is missing, and some strange fables are spun. The most outlandish one is that Möbius and Listing discovered the strip "together", and "decided" to give priority to Möbius. Another, from an old popular book that I can not recall, is that Möbius was inspired by the shape of a twisted stock garter dropped by his maid.
The Möbius strip is not exactly a polyhedron, what is the connection? Möbius worked in Leipzig, and Listing in Göttingen. Listing was not exactly unknown, he was recommended for the Göttingen position by Gauss himself. Did they both work on the Académie problem? were they aware of each other's work? did they correspond or meet? why did the name "Möbius strip" stuck? Can somebody fill in the context?