Suppose: a = b

It then follows that
a * a = a * b
a^2 = a * b
a^2 - b^2 = a * b - b^2
(a - b) * (a + b) = (a - b) * b
a + b = b
a + a = a
2 * a = 1 * a
2 = 1

Is this derivation really correct? Or is there somewhere a mistake? If so, where?

Of course the derivation should be wrong. 2 is not equal to 1. The mistake is in step

(a - b) * (a + b) = (a - b) * b

to

a + b = b

This step is not allowed if a = b. Because then you are dividing by zero. And dividing by zero is nonsense.