Insofar as the intuitions in intuitionism are intentional states, then yes, I'd expect a closer to one-to-one correspondence between the numbers of possible intuitions and of possible intentional states. I should also agree when you say that realists do not end up with that correspondence, but I will also go into a little detail about why those realists end up where they do.
Not, however, then, that this is an absolutely universal explanation, but: in model theory, there are some juxtapositions where you can have a model, as a set, be only countably infinite, and yet it "logically" covers uncountably many objects. More broadly, the Löwenheim/Skolem number of a logic, say, might be so large that you have an uncountable base, yet upwards-wise you can still push the model-theoretic functions to "cover" more objects than are enumerable modulo the relevant base. Insofar as models are semantic structures that are about proper classes of objects, it seems possible to have a mismatch between the number of intentional (semantic-aboutness) states in a system, and the number of objects proposed by the system.
A similar moment occurs in finitely-axiomatized proper-class theorizing, except here we can have a relatively finite model-theoretic set yet "cover" (or, more accurately, "refer to") an absolutely infinite one. Hypothetically, you can use one class axiom (an axiom for a specific class) to comprehend class-many objects. (For some reason, having one axiom scheme with arbitrarily many instances is not entirely the same thing; but it is comparable thereto.) So perhaps you might think of this as having a single intentional state comprehending plural objects; and eventually, we'll find ourselves in the dark caverns of the unrestricted quantifiers, wondering if our flashlights' reach is as far as possible, or if we can't quantify over "absolutely everything," esp. not by a single symbol! (So then not just bare unrestricted quantification, but internally plural quantification, are opened up as questions, i.e. do we refer by a single term to {the category of all sets} or directly, but then plurally, just to {all sets}?)