I don't understand the problem. x and y are free, so you can choose x=y. Then obviously, f(x)=f(y). Then it should follow x=x xor c, thus c=0. Or did they forget to claim that x and y are different?
If c=0, then it says that f(x)=f(y) if and only if (<=>) x=y. In other words, for all x≠y, f(x)≠f(y). Thus the function is 1-to-1. This is a nontrivial property that is difficult to check.
It’s easy to do. Einstein did the same thing with E=mc2.
Equality is a matter of identity. Equivalence is a matter of behavior.
The speed of light is not an absolute constant as Einstein believed. C just represented the speed that light can travel as a relationship between energy and mass.
My favorite form of the equation is C equals the square root of energy divided by mass.
Behaviorally all that means is that as the energy to mass ratio goes up, the speed of light goes up. And as the mass to energy ratio goes up, the speed of light goes down. Hence time dilation, black holes, etc.
That isn't what's happening in relativity. That might be a neat trick for you to remember whether an effect is dilation or contraction (though I personally find it more confusing), but the speed of light does not change in different frames of reference -- this is a fundamental property of relativity (and is an assertion by Einstein, who argued that the speed of light derived from Maxwell's equations means that if light functions in all frames of reference it must propagate at the same speed).
And while the mass-energy equivalence equation you mentioned is used in the way you described (rearranging it to have the speed of light as one term), it's actually used explicitly because the value must be constant -- the implication being that mass-energy is an invariant in relativity (though technically this derivation is backwards -- E=mc² is the conclusion you reach after assuming mass-energy conservation).
Relativity is already unintuitive enough, I personally find that adding incorrect-but-seemingly-intuitive explanations probably hurts your understanding more than it helps.
The full equation is:
E = mc^2 / sqrt(1 - v^2 / c^2)
This is coordinate/frame-dependent since it depends on speed. Here, m = mass of the particle which is also the same in every reference frame. You could define an effective mass as m' = m / sqrt(1 - v^2 / c^2) but that obscures the point imo.
To address your point directly, when you say "as the energy to mass ratio goes up, the speed of light goes up", that is just not true both mathematically and physically.