I would say it's the opposite. A unit conversion is not fundamental at all, but a matter of arbitrary convention. 0C and 32F are exactly the same temperature. Mass and energy are exactly the same property. c^2 is a constant, so E doesn't even vary as the square of anything - it's just a linear unit conversion. If you follow it further, you find c is only even in there because some other units are defined in terms of it. That is, the ME equivalence doesn't really have anything to do with the speed of information propagation per se - but our chosen unit conventions do.

It's because of people's fondness for legacy and backwards compatibility that we don't use natural units all the time, and still write "e=mc^2". I find this insane, personally, but that's just the way it goes.

I guess I should add that apparently this is still controversial? But in my mind, the only reason for this is that people find meaning in units. In this case, ME being the same property implies duration being exactly the same as length, which implies spacetime is a unified, regular 4D manifold. ME not being the same property implies time is special. I've been fully persuaded that duration is fundamentally exactly the same thing as length, so I've chosen my side, in case there's any serious controversy.