No you're totally right! In fact I think you're even more right than you said in the original post: the analogy you draw isn't just an analogy, they are in fact

No you're totally right! In fact I think you're even more right than you said in the original post: the analogy you draw isn't just an analogy, they are in fact the exact same thing. A class is (amongst other things) a module, and a closure, and a type. In functional programming these things are (usually!) deconstructed rather than all being facets of the ‘class’ concept, but they're all still there and they all work exactly the same way as in OOP (perhaps modulo mutability, depending on choice of languages — but you can see local mutability as syntax sugar over function calls anyway).