This is false, and in a meaningful way. Energy and momentum are not the same. Momentum scales linearly with speed, whilst momentum scales quadratically. One case where this difference matters a lot is the Oberth effect, where the fact that Delta V is a measure of speed rather than energy is of key importance.
The Oberth effect states: "the most energy-efficient method for a spacecraft to burn its fuel is at the lowest possible orbital periapsis". Or, more accurately "the use of a reaction engine at higher speeds generates a greater change in mechanical energy than its use at lower speeds". It just turns happens to be that, in orbit, you are going fastest at the lowest point. This effect is a direct result of the scaling differences. Adding 1m/s will always require the same change in momentum, but the faster you are going, the more it will increase your kinetic energy. For a 2kg object, going from 0m/s to 1m/s increases kinetic energy by 1 joules. Going from 1000m/s to 1001m/s increases kinetic energy by 2001 joules. It is key here that, when expending reaction mass at a given exhaust velocity, it will always result in the same change in momentum, whilst resulting in a different change in kinetic energy. Delta-V is just the total speed change that can be achieved by all momentum generated through reaction mass. The rocket equation is about converting this momentum into a speed by compensating for the fact that a rocket gets lighter as it expends reaction mass, to get to Delta-V.
If Delta V was a measure of energy, then it shouldn't matter where you use your delta V, you always get the same kinetic energy back. Luckily Delta V is a measure of momentum, which allows us to exploit the Oberth effect for more efficient orbital changes.
As for exhaust velocity, _if_ your fuel is your energy source, the exhaust velocity and the energy density are indeed related. But consider an ion engine, where the reaction mass is just mass, and the energy comes from solar panels (or another source of energy).