Reduction to proof undecidability of the problem: machine M and N accept infinitely many words
Here is the reduction from the halting problem, whether a given TM halts on empty input, as given in Yuval's comment.

For a TM $M$, let $K$ be the same as $K$ but ignoring inputs. That is, for any input, $K$ will first erase the input so that it looks like an empty input is given. Then $K$ will simulate $M$ on empty input. When the simulation halts, $K$ accepts. Now consider whether $\langle K\rangle\#\langle K\rangle$ is in the language specified in the question.

Exercise. Construction a reduction from the other halting problem, whether a given TM halts on a given input $x$.