If you express it in a different unit system all E, m, and c will take different numerical values, but the relationship will still be true.

So there’s nothing special about the metric system. When we want to discuss this kind of relationship without reference to human convention we talk about a quantity’s dimension (not geometric like 3D). A Meter and a foot both have dimensions of length. c has units of length/time. 1 kilogram and 1 gram both have units of mass. And so on.

https://en.wikipedia.org/wiki/Dimensional_analysis

Kilograms and joules have been interchangable (well, they've had a defined relationship) for a long time though, you can calculate work in a completely Newtonian context and see that the energy requires is linearly related to the mass and quadratically to velocity (like with a spring).

That the direct conversion from mass to energy follows the same shape isn't really surprising, it sort of has to.

That said, the joule was only explicitly defined as kgm/s^2 in 1946 (or 1935), after Einstein and nuclear physics.

The kinetic energy of a body, in purely Newtonian mechanics, is Ec = (m × v²) / 2. Since energy is measured in J, mass in kg, and speed in m/s, it follows nicely that J = kg × m² / s². Or, we can also take the potential energy of a body at some height h above the Earth: Ep = m × h × g, where g is the gravitational acceleration, ~9.8m/s², and we get the same units.

It's really not in any way surprising - this is basically the definition of the Joule. The link between units of kinetic energy and units of thermal energy is actually more surprising.

The surprising thing about E=mc² is that it gives a definition of energy for a completely stationary body outside any external field.

One way to look at it is actually that this is simply the kinetic energy of the body, and that all "stationary" bodies are in fact moving with speed c on the t coordinate in 4D space-time ("a body which is not moving in space at all is moving with speed c towards the future"). [Note that of course speeds are all relative to some reference frame.]