Yes, there is an often overlooked unit 1 at the heart of dimensional analysis.
It counts discrete things that combine as integers, but are not convertible (commensurate) between different instances of the unit, like apples and oranges.
A hen lays 3 eggs a week (1egg/T), and a car factory makes 4,000 cars a week (1car/T). Dimensional analysis says they are both 1/T, but we know they are really not commensurable.
P.S. Things that do not combine as integers: water drops (1+1=1); rabbits (1+1=2^t for some time constant t).
It counts discrete things that combine as integers, but are not convertible (commensurate) between different instances of the unit, like apples and oranges.
A hen lays 3 eggs a week (1egg/T), and a car factory makes 4,000 cars a week (1car/T). Dimensional analysis says they are both 1/T, but we know they are really not commensurable.
P.S. Things that do not combine as integers: water drops (1+1=1); rabbits (1+1=2^t for some time constant t).