Let's rewrite the given condition with different letters...
f(a + b) = f(a) + f(b) for ALL/ANY a,b that you choose to be real numbers.
So f(2 + 3) = f(2) + f(3) = f(1) + f(4) = f(1 + 4)
You're confusing independence and the letters x, y.
Since "0" and "0" are both real numbers, we are justified is considering what happens when we let a = 0 and b = 0.
f(0 + 0) = f(0) + f(0)
f(0) = 2f(0)
f(0) = 0
a and b are independent, because we choose them, independently (!), before invoking the function.
In your example, y = f(x). In the given problem, y is an input, not a function value.