Probability of iid random variables to be equal?
No, your calculation for the continuous case is wrong. It should be $ P(X_1 = X_2) = \displaystyle\iint_D f_{X_1}(x) f_{X_2}(y)\ dx \ dy$, where $D = \{(x,y) \in {\mathbb R}^2: x = y\}$. But $D$ has two-dimensional measure (i.e. area) $0$, so the answer is $0$.