One thing about 6/8 that is not literally communicated by the notation is the rhythm: the first and fourth beats of the bar are stressed. But not equally, because that would be indistinguishable from 3/8.
And 6/8 says nothing about the beat structure within the 6 beats. You're stating the most common occurrence, but that's not the definition of 6/8.
The point stands that time signatures are not rational numbers (as you acknowledge, the unit of the denumerator can be scaled) so 3/4 and 6/8 aren't part of some equivalence class. They have different meanings which are only sustained by the presence of a cyclical pattern of stressed notes. So if there's some music theory you can link to which elaborates on all the possible different rhythmic patterns of 6/8 vs 3/4, that would provide examples of the distinction we both agree exists.