All prefixes with same length as their suffix is a regular language
Given a DFA $A$ for $L$, construct an NFA which operates as follows:

The NFA starts by guessing a state $q$ which will be the state that $A$ is on after reading $x$.
The NFA will maintain two states, $q_1,q_2$, the first initialized at the initial state of $A$, the second initialized at $q$.
For each symbol $\sigma$ read, the NFA guesses a new symbol $\tau$ (which corresponds to a symbol in $y$), and advance $q_1$ using $\sigma$ and $q_2$ using $\tau$.
The NFA is at an accepting case if $q_1 = q$ and $q_2$ is an accepting state of $A$.