The von Neumann construction of natural numbers is a mechanic rule to create finite length symbols interpreted as natural numbers. We use set membership to get ordinality and also the successor relation.
To extend that thinking to fractions, we need a mechanical rule to deal with finite lists of symbols that represent a fraction. Ordered pairs in lowest denominator is a way to approach this.
The unit fractions are either an approach or a partial test bed for considering such ideas.
We can simply take the unit fractions as given. We can then define a fraction of the same denominator using the von Neumann construction on the numerator holding the denominator fixed.
This falls short of what we would like in this approach.
We would like an ordinal set (2/3)[set] to include fractions like 1/2 in some way. But is this including 1/2 as a set or as a simple pair of numbers?
We could define ordered pairs of numbers, build some sets out of those, and then further have sets composed of those and possibly of themselves. So we can do a theory of types or bootstrapping approach to fractional sets as ordinals.