A student claims that x=2 is the only solution of the equation x/x-2=2/x-2. Is the student correct?

The way you have those slashes there are a bit confusing.

If what you're asking is the same as x (x-2) = 2 (x-2) than yes, because you would just divide both sides by (x-2), since everything within parentheses acts as a singular unit. It'd be like saying x (9) = 2 (9)

Multiply each denominator by the other terms numerator.

This gives us x (x-2) = 2 (x-2), which can be expanded to x^2 - 2x = 2x - 4

If we move everything to one side we get the quadratic equation x^2 - 4x+4=0

This can be factored into (x-2) (x-2) = 0

Set each term equal to zero and solve

x-2=0 so x = 2, the terms are the same so only x=2 is true so the student is correct.