Question

Two squares are chosen at random on a chessboard the probability that they are adjacent and touching each other diagonally

Answer 1

assumes that the squares have "identities", and that thus that switching the squares is considered a separate outcome. If you're not interested in that, just divide my answer by 2.

Sampling with replacement:
4.7852%4.7852%
Sampling without replacement:
4.8611%4.8611%

We pick the first square randomly:
4 / 64 chance it's a corner
24 / 64 chance it's a side