Your example seems to illustrate a different construction. Subject (nom.) + хотеть + direct object (acc.) is a familiar pattern from previous exercises. But this sentence is different. It appears to be Indirect object (dat.) + хотеться + subject (?). If the milk isn't the subject, as per your below comment, what part of speech is it?
No, it's not a subject. The milk does not want itself. The explicit subject is missing in this construction - this is hard to imagine in English, but not so uncommon even in languages close to English, such as German. Since you study it as well, you may recognise examples like "Mir ist kalt". (The subject - "es" ("it") is implicit.)
Even in English it's not so hard, just a little more removed in thinking: "It would be nice for the car to have" = "The cat would like [to have]....", or "one wants for the cat (to have) some milk".
It's not normal English, but if you read some 19th Century English literature. you can come close to the Russian in intent.
Because the cat is very polite and well-mannered, and never "wants" anything, but evermore "would like" something - or, more precisely, is in such a social position that the noble beast assumes - rightly so - that "one wants for the cat some milk" - "one" being whoever is there is wait hand-and-foot on the deserving animal. Such is a cat's existence...
We learned earlier that to express “the cat wants milk” it’s just “кошка хочет молоко” - what is this “хочется”? Where is this “-ся” suffix coming from? In what contexts is it necessary? I don’t understand the purpose of this extra complication. What would its English equivalent be?
-ся (or -сь if the word ends with a vowel) is the suffix that makes the verb reflexive. In the case of хотеть, making the verb reflexive gives it an impersonal or passive flavour, so it's a little less direct and more polite. That's why above they've translated it to "would like", which is more polite than "wants".
Partly for the same reason it can't be "The cat wishes some milk", which is given for me as the Correct Answer: because in English "wishing X" is just not a thing. Wishing for? Sure. Not just plain wishing. Unless the subject has psychic powers and is "wishing X into existence." But that's an entirely different story.
It was there by mistake, I've removed "wishes" from the list of acceptable translations.
Btw, you can "wish something" in English, but that something has to preceded by an indirect object: I wish you luck.
Given my first point - "wishes" is an inaccurate translation of the verb "хочется", the article "a" will not fit into a more accurate translation. "Хочется" is the verb that is used to describe a current state of a subject. So unless you are absolutely certain that any given cat presently wants milk, "a" would be a poor article choice.
They are not that different; I can't think of a situation where I could use one but not the other. My personal preference (perhaps not universally shared) is to use "мне хочется" to describe my whims or momentary desires (Мне хочется воды) and "я хочу" to describe more sustained and meaningful wishes/desires (Я хочу поехать в Перу).