I don't think "homomorphism" is quite the right word here.

The map from dimensional quantities to units is structure-preserving, so yes, it is a homomorphism between something like rings. For example, all distances in SI are mapped into the element "meter", and all time intervals into the element "second". Addition and subtraction is trivial under the map (e.g. m+m=m), and so is multiplication by a dimensionless quantity, while multiplication and division by a dimensional quantity generates new elements (e.g. meter per second).

Converting between different measurement systems (e.g. SI and CGS) adds various scale factors, thus enlarging the codomain of the map.