To my understanding, Utilitarianism is just the principle of maximising utility, where utility is defined as the state of happiness. However, the name itself does not seem to specify that this utility is specifically happiness. Could not the philosophy be applied to other objectives? And if so, would it still be referred to as Utilitarianism?
Not necessarily; there is an "historical" link between the two points of view.
Compare Utilitarianism :
Though there are many varieties of the view discussed, utilitarianism is generally held to be the view that the morally right action is the action that produces the most good. There are many ways to spell out this general claim. One thing to note is that the theory is a form of consequentialism: the right action is understood entirely in terms of consequences produced. What distinguishes utilitarianism from egoism has to do with the scope of the relevant consequences. On the utilitarian view one ought to maximize the overall good — that is, consider the good of others as well as one's own good.
with Hedonism :
Ethical or evaluative hedonism claims that only pleasure has worth or value and only pain or displeasure has disvalue or the opposite of worth. Jeremy Bentham asserted both psychological and ethical hedonism with the first two sentences of his book An Introduction to the Principles of Morals and Legislation (1823): “Nature has placed mankind under the governance of two sovereign masters, pain, and pleasure. It is for them alone to point out what we ought to do, as well as to determine what we shall do”.