I thought "units canceled out" was the entire point. The units we use say everything about our assumptions such as the definition of distance and time (concepts, for example, closely tied to our understanding of light) and dimensionless constants allow us to safetly hide some of those assumptions.

Philosophically this is significant because these constants provide us our most basic hypothetical methods of communication with other technologically advanced species in the future, provided they share a relatively similar linear number system (which can be based on ratio of protons between different elements).