Because the name Quincas Borba belongs to two characters: the philosopher who invents the doctrine of Humanitas, and the dog he names after himself. The book does not say which of the two gives the title, and it is the narrator himself who raises the question in the last chapter and leaves it without an answer.
The philosopher explains the dog's name to Rubião for two reasons, "one doctrinal, the other personal." The doctrinal one is that Humanitas dwells everywhere and exists in the dog too, which "can thus receive a human name." The personal one is survival: "If I die first, as I presume, I shall survive in the name of my good dog." When he leaves for Rio, the philosopher weeps for the dog, and the text puts it this way: "Quincas Borba wept for the other Quincas Borba."
The philosopher dies early in the book, and the dog stays with Rubião to the end. More than once Rubião supposes the two are one: "it occurred to him that the two Quincas Borbas might be the same creature, by the effect of the dead man's soul entering the dog's body." Later he suspects that a voice that advises him comes from the dog or from the philosopher's spirit, an idea he dismisses as absurd, but, just in case, he scratches the dog's ears.
In the last chapter, when recounting the dog's death, the narrator says that the reader would probably ask "whether he, or his deceased namesake, is the one who gives the book its title, and why one rather than the other," and answers that it is a "question pregnant with questions, which would lead us far." The protagonist, Rubião, does not give his name to the book.