The way I see it, should-wise: issorted should simply check whether lt holds for each consecutive pair. sort should have an output for which issorted is true, if possible. The default lt should be <=. sort([1, 1, 2]; lt=isless) should error out, because it can’t be done, just like sort([1, im]; lt=isless) can’t be done.
Implementation wise, the possibility of using arbitrary comparisons means it can’t be entirely specialized on type (like above), and it’s not even guaranteed there is an algorithm, consider lt=(x, y)->false. But the domain of lt for sort is a strict subset of that for issorted.