The Maximum Height A Block Can Climb On A Ramp

You're on the right track since you think of using conservaion of energy and momentum. Velocity of the block of mass $m$ and and ramp $M$ are taken the same in horizontal direction since the system moves together. Picture-wise, it's like in the inelastic collisions, when two masses move together with the same velocity after the collision.

Or, you may think in the frame which the block $M$ is fixed. The block $m$ is moving up while surffering gravitational force, $- m g hat{y}$, and the inertial force from the acceleration frame of $M$, $- m a_M hat{x}$.

Though, the forces become a little complicated, but eventually, the block $m$ will be tired of climbing, then stop for a short rest. It will starting moving backward (and getting lower) in the next moment. Behold, at this highest posiiton, in this very moment, $m$ is stationery at the $M$ frame. What is that meaning "stationery"? No doubts, they are moving together with a same velocity.