As you note, technically there's a standard but artificial trick for regimenting the natural many-sorted language of mathematicians into a single sorted formal language. Now, there's always a trade-off in regimentations between sticking more closely to the logical shape of the discourse we are regimenting and getting a slicker formal calculus which is easier to theorize about. And in this case, for many purposes, the price is right -- a certain cost in readability buys us a formalism that appeals to logicians' appetite for clean Bauhaus lines.
But, some would say, we shouldn't let the availability of the trick lead us astray. Some working mathematicians (round my neck of the woods) take their mathematical universes to be strongly typed -- real numbers (say) are one sort of thing, vectors (say) are another sort of thing, and sets (say) are yet another sort of thing. You can model or represent numbers and vectors as sets, but it's a kind of pointless philosophical nonsense (they'd say) to ask whether numbers really are sets, for example. The numbers and the sets belong to different mathematical universes. Conceptually speaking, these mathematicians might say, the trouble about regimenting talk about reals and vectors and sets as talk of objects belonging to One Big Domain is that it encourages some nonsense questions about which objects in the domain are identical to other objects alongside the sensible questions. So, conceptually, it is better to stick with the everyday mathematicians' sorted talk. Or so the story goes. (I'm not endorsing it, just gesturing to why it might be wondered whether the technical trick is merely technical.)