I’ve heard native speakers use “half” but never “halves” in such sentences. If we cut an apple into two identical pieces, we get two halves. It shouldn’t, therefore, sound wrong if one used “halves”. Would it be wrong to use the plural here?
It would be “please cut the apple into halves”, which is perfectly correct.
But you are right, in practice it is rarer in everyday usage than “please cut the apple in half”.
“To cut in half” is a fixed expression which stands for the verb “to halve” (which is more formal or archaic) and describes the action of dividing into equal parts. There is no equivalent phrase for other proportions. For example a budget or workforce can be “cut in half”: it has a similar function to decimate (though a different proportion is involved).
By contrast, “to cut into halves” focuses on the end product, — the halves — and the means (to cut). It is a living phrase and other proportions and mechanisms can be substituted at will (“cut into quarters”, etc).