Say we roll n identical, fair dice, each with d sides. (Every side comes up with the same probability.) On each die, the sides are numbered from $1$ to d with no repeating numbers, as you would expect. So it's an ordinary d sided die pool.
How would we calculate the odds of the highest rolled die value from a given dice pool equaling the highest rolled die value from a different dice pool?
Let $(X_i)_{i=1}^n, (Y_i)_{i=1}^n$ be the sequence of results from the individual die, which are mutually-independent and identically uniformly-discrete-distributed random values.
So, to get you on your way:
$$\begin{align}&\qquad\mathsf P(\max{(X_i)}_{i=1}^d=\max{(Y_i)}_{i=1}^d) \\[1ex]&=~ \sum_{k=1}^d \mathsf P(\max{(X_i)}_{i=1}^d=k)\cdot\mathsf P(\max{(Y_i)}_{i=1}^d=k)\\[1ex] &=~\sum_{k=1}^d \mathsf P\left(\bigcap_{i=1}^d \{X_i\leq k\} \smallsetminus\bigcap_{j=1}^d \{X_j\leq (k-1)\}\right)\cdotp\mathsf P\left(\bigcap_{i=1}^d \{Y_i\leq k\} \smallsetminus\bigcap_{j=1}^d \{Y_j\leq (k-1)\})\right)\\[1ex] &=~\sum_{k=1}^d \left(\mathsf P(X_1\leq k)^d -\mathsf P(X_1\leq (k-1))^d\right)\cdotp\left(\mathsf P(Y_1\leq k)^d -\mathsf P(Y_1\leq (k-1))^d\right)\\[1ex] &=~\sum_{k=1}^d \dfrac{(k^n-(k-1)^n)^2}{d^{2n}}\\[1ex]&~\ddots\end{align}$$
