There's a really important thing going on here.
Like Tom10 said, there are more differences than similarities. For example, if you travel alongside a sound wave, everything is fine. Well, you get sonic booms, but you can still do it. If you try to travel alongside an electromagnetic wave, you run into problems with relativity. You just see the wave get redshifted more and more while it travels in the same direction at the same speed.
Compare this to other types of waves. Sound waves have a maximum frequency due to atomic details. And different modes of propagation. There are weird nonlinearities. And if the amplitude is too high, your solid will break or melt. You have $10^{23}$ particles bouncing around, and there are a lot of different phenomena!
Similarly, surface waves on water have lots of ugly properties. They can break, they have oddly shaped patterns called cnoidal waves, and of course, if you get a cup of water and pour it on the ground, there's not much wave behavior at all...
And finally, heat! I think by "heat waves" you mean precisely "infrared radiation", and by "light waves" you mean precisely "visible radiation". Regular, run-of-the-mill "heat" is basically just sound. Vibrations of atoms. (Interestingly enough, there's also a quantization that comes about for sound, and you can use the quantum mechanics of quantum sound to calculate thermal properties of matter.)
You might ask why so many different phenomena (waves on the surface of water, sound waves, electromagnetic waves, and even gravity waves) can be described by the same notion of a "wave" in some approximation. (Electromagnetic radiation is the best approximation of an ideal wave in that list, by the way!) The reason behind this is that somehow, some way, classical physics only needs to talk about accelerations. And when you deal with approximations involving accelerations (small amplitude sound waves, small amplitude surface waves on water, small amplitude gravity waves), it's very hard to wind up with anything other than a wave equation. (Related shameless self-plug: "Why are sine/cosine always used to describe oscillations?")
The real answer for your daughter, is that sound waves are an emergent property when you have $10^{XX}$ particles bouncing around, whereas electromagnetic waves are a property of space itself. You can say there is a sort of medium through which waves propagate. That medium is space. Space has properties. It has an energy density from electromagnetism. It has an energy density from gravitation. General relativity tells you energy density=mass, so space has mass too. It's a medium. Just a bit of a weird medium which obeys Lorentz invariance, and Lorentz invariance is a bit counterintuitive for our hunter-gatherer minds, so a lot of physicists/teachers prefer to tell people it's "not a medium". All they mean to say is that it's not a medium in the same way that a block of metal is a medium.
Maybe it helps to imagine this: Imagine somewhere deep in outer space. Maybe you imagine a big box a meter cubed, and the only thing inside it is a single hydrogen atom. Sound can't propagate, there's nothing for the hydrogen atom to hit to pass its energy on to. Now imagine there's a light wave passing through the box. Well, what is light? Maxwell teaches us it's just part of the electromagnetic field, and electric fields accelerate protons and electrons. If you looked closely, you'd see the proton being pulled one direction and the electron being pulled the opposite direction. The whole thing vibrates periodically. An electromagnetic wave is propagating through empty space.
If you fill the box up with hydrogen atoms, space is still there, so the electromagnetic wave can propagate just fine (although it's effected by the matter creating its own electromagnetic waves in response. Matter can react in a lot of different ways so that can be a very complicated situation).
