This is exactly correct. And once you think of it as the variable, you can then make the leap to thinking about it as dimensions, which has some nice properties

This is exactly correct. And once you think of it as the variable, you can then make the leap to thinking about it as dimensions, which has some nice properties in physics i.e. vector addition/subtraction leads to tuples of dimensions, while the product doesn't - which explains why we can't add/subtract different units, but we can multiple them together.