In fact, in automata theory (which departs a lots from the origins of Kleene, Rabin and Scott), there are many forms of automata that are not finite. This arises for several reasons.
Pushdown automata, for instance, are automata that have an infinite set of configurations (these have a finite number of states, but the reality is that these should be thought as 'infinite automata').
In the same vein, there ar other examples of infinite automata for which the state space is infinite, but with a lot of structure. For instance one considers the class of automata that have as state space a (finite dimension) vector space, and as transition functions linear maps (plus some initial an final things). These are known as weighted automata over a base field (due to Schützenberger in 61). These can be minimized and tested for equality. Other examples include register automata (these automata have a finite set of registers, and work over an infinite alphabet: these can compare letters with registers and store letters in registers), and the more modern form of nominal automata (that have the same expressiveness, but have better foundations and properties). Emptiness of such automata is decidable.
But, even for studying finite state automata, it makes sense to talk about infinite automata. Indeed, consider the category of finite state deterministic automata that accept a fixed given language L, equipped with the standard notion of automata morphism, then this category fails to have an initial and a final object. However, if you live in the category of all deterministic (say countable) automata, then there is an initial object (the automaton that has as states A∗, as initial state the empty word, when it reads letter a in state u it goes to state ua, and a state is accepting if it belongs to L). There is also a final object (that has as states languages!). The existence of these two objects are one way to explain at high level why deterministic automata can be minimised and is tightly linked to the Myhill-Nerode congruence.
To conclude, there are infinite automata, but the models that are first studied in a lecture are always the finite state ones.