What is being spoken about previous to this exchange between the protagonist, Bella, and Jacob, is how warm Jacob feels to the touch. Jacob shrugs this away as a "wolf thing" because Meyer explains in New Moon that while vampires are abnormally cold to the touch--like stone or marble--werewolves, even in human form, run much hotter than a human could. Meyer says in the book that Jacob's body temperature runs between 108-109 degrees farenheit, about 10-11 degrees hotter than a normal person. Because of this, Jacob is always hot to the touch. This is why he says being that warm is a "wolf thing" because it is a physiological part of him now.
When Bella counters with "No, it's a Jacob thing," she is not referring to the physiological side of Jacob but rather the emotional one. Jacob is described in New Moon by Meyers as happy, cheerful, and compassionate. These are traits that are encompassed under the umbrella of saying someone is "warm." In this sense, to be a "warm person" means to make others feel happy and good about themselves. So, while Jacob takes his body temperature to be a purely physical thing, Bella uses it to refer to his "warm" personality as well.