It seems to me that the "dimension" of a measurable quantity answers the
question "What are we measuring?"; a question of KIND or CONCEPT - often
based on some conceptual model of the phenomenon under consideration and
its interaction with the measuring instrument.
A "unit" is attached to a mathematical entity (a signed number in the
case of a scalar quantity) to specify the SIZE or AMOUNT of the
measurement result, in terms of a chosen standard amount.
I would classify length, mass, time, angular opening, charge, etc as
"dimensions"; and meters, kilograms, seconds, radians, and coulombs as
"units". The radian is not dimensionless anymore than the centimeter is.
They are each the unit of a dimension (angular opening and linear
extent). The fact that the size of an angular opening can be
measured/defined (in radians) by dividing two lengths is neither
bothersome nor unique.