Besides costume mustaches that people put on their cats, there are currently two different concepts of cat mustaches. Cats occasionally have natural facial coloring that looks like a man's mustache. Also, people take photographs with their cats that create the illusion of a mustache on the person.
For humorous reasons, a person sometimes poses with his cat positioned in front of his own face with the cat looking directly upwards. When the cat's eyes, ears and forehead are hidden, and the cat's small nose is lined up with the person's, it obscures the person's lower features. This creates the illusion that the person has a handlebar- or imperial-style mustache and beard.
Additionally, there is a Canadian cat named Stache who was born in 2012 who happens to have a mostly white face with a distinctive black marking just under his nose that looks just like a mustache.