This presence of cat-ness in Llewyn’s very name matters because two of the chief themes Inside Llewyn Davis explores are identity and authenticity.
Llewyn desperately needs to find himself. He is terrified of the mere “existence” of his father and sister, and craves success—affirmation, really—as a folk singer. But if any of his contemporaries catch a break, he scorns them. He mocks Jean’s dream of having a family life as careerist, square, and sad. He’s thankful for the gig, but can’t believe Jim wrote something as banal as his song “Please, Mr. Kennedy.” The Irish folk group based on The Clancy Brothers? Well, their sweaters are nice. He’s a penniless couch-hopper, but screams at moussaka chef extraordinaire Lillian Gorfein that music is “how I pay the fucking rent!”
These identity problems show up all over the dialogue, especially with the amount of time spent figuring out names. Al Cody is really the less- sexily named Arthur Milgrum. A beat poet driver Llewyn encounters goes by the surely made up name of “Johnny Five.” The musician Roland Turner mishears Llewyn’s name as “Lou N. Davis.”
The name and identity of the cat is mysterious, too. Llewyn is always caught stammering saying the cat is his, or the Gorfeins’, or that it’s a he or a she. He asks the cat several times for its name (to no response).
The focus on names fits right into the theme of identity. In folklore and mythology, one of the unbreakable laws is that names have power. To control anything of magic, you must know its true name. And cats in particular have universally been portrayed as nearly impossible to control.
James Joyce’s influence is recognizable throughout the movie (i.e., Llewyn’s epic journey stays mostly within a few square miles, just like Leopold Bloom’s in Ulysses), but it’s a work by T.S. Eliot that perfectly illustrates the odd relationship between cats and identity Inside Llewyn Davis embodies. His poem The Naming of Cats asserts that a cat in meditation is always “engaged in a rapt contemplation / of the thought, of the thought, of the thought of his name.
This theory that the cat is an extension of Llewyn also helps put the ending of the movie in context. When Llewyn leaves the Gorfeins’ for the “second” time in the final scenes of the film, he keeps the cat inside. This comes after he’s finally learned its name: Ulysses. By doing so, I think the uncontrollable, unpredictable Llewyn also comes to terms with a part of himself. He has been awoken from the dream that he’s an undiscovered genius, and from the erroneous notion that talent exists in a vacuum—that any of his poor decisions and arrogant assholery wouldn’t somehow limit his success.
He still has to pay the price for his behavior (most notably in the form of a back alley beating), and he still has a long journey ahead. But Llewyn has reconciled with the cat—learning a name that, like Llewyn says of a beloved folk ballad, "was never new and won’t get old." And that makes all the difference.