The physical kind is a point in an N-dimensional rational space of exponents of base units, such as Mass * Distance ^2 * Time^-2 or (1,2,-2) for energy (maybe with some carry over ^0s for unused slots of a >3D meta-type space). Additive (like add, subtract, compare, and convert) are only meaningful at the same point in that lattice. If you are interested in learning more, this is all a part of dimensional analysis: https://en.wikipedia.org/wiki/Dimensional_analysis .
Even the dimensionality of that rational exponent space (the "unit system", if you will) is more of convenience / convention than fundamental. https://en.wikipedia.org/wiki/Planck_units clobber most things away leaving only 1 dimension (and a tendency for complex rational exponents). Meanwhile, the SI unit system is very "dimension promiscuous" (mass, length, time, electric current, temperature, amount of substance, luminous intensity) [1] to avoid non-integer exponents (but you cannot really forever since as soon as some formula has some square some solution of it has some root that probably gets you a 1/2 exponent).
There are some situations with not purely "conversion factor" scale offsets like thermal units (e.g. Celsius to Fahrenheit) that people often describe in math-ese as "affine conversions". Something like GNU Units attends these things, but they are kind of "conversion only" orphan step-children more than "real" types which "compose better". In physics formulae one will often only care about delta-Temperature, not levels, for example.
EDIT: The implicit creation of a potentially brand new type any time you multiply | divide two extant things is, incidentally, why you need either a dynamic or a very powerful static type system to express these things.
[1] https://en.wikipedia.org/wiki/International_System_of_Units