A is what you mean. B means that there are two cats. The first cat said something like: "Fatty, come over here!" - it called Fatty to join it. Sometimes you see B used as a subject in a sentence e.g. "The cat (who is) called Fatty purred and drank some milk."

'A' is a complete sentence, whereas 'B' is not (or only when given as a short response to a question such as 'Which cat?, as Michael points out). The verb needs to be in continuous form, as a name is given not as a moment in time, but as a continuous thing. Therefore, 'is called' and 'was called' are acceptable, but simply 'called' is not. You would say 'was called' if the cat is deceased or if you are telling a story about a cat that you interacted with in the past but don't anymore (i.e. "the cat which belonged to my first housemate was called 'Tom'"). Alternatively, you may say "The cat called Fatty ..." followed by a verb, in a Subject-Verb(-Object) structure. E.g. The cat called Fatty eats too much.

A says the cat's present name and B says the cat's past name that maybe it is changed now or not