Astudent 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.