Yes, there is the so-called impedance of the medium which is the ratio of the magnitude of the electric field divided by the magnetic field. And in some media there will be a phase shift between the two.
EM waves don’t travel through media in which the apparent speed of light is less than C. They turn into electromechanical motions involving the positions of the electrons. This also explains Brewster’s angle and the polarizing effect of angled reflections off of glass surfaces. So, light enters one side of a window and exits the other side, but it isn’t just electromagnetic radiation while it’s inside the window. That’s why the speed is reduced.
What’s meant by the speed of electricity traveling through a wire is actually not well defined. If you mean the ability to transmit signals or power, the marble analogy is pretty good, except 1) in marbles the speed is determined by the speed of sound through glass, and 2) in electricity the speed is determined by electromagnetic and electrostatic interactions around the conductor and the relaxation time for electrons themselves (which also gives rise to the “skin effect”. Coaxial cables are designed in part to maximize this speed but it is rarely more than 1/2 to 2/3 of c (like an earlier post pointed out).
If you mean the speed of the electrons, the earlier mentions of drift velocity are correct, except that the values discussed are more appropriate for a wire carrying a typical safe maximum current or a significant fraction of it. In situations like telephone wires or doorbell wires or a heavy extension cord feeding a small lamp, where the current is nowhere near the maximum safe rating, the drift velocity is proportionally smaller. In situations like the leads on your voltmeter, which has a high impedance and therefore a low current, the drift velocity could be a million or a billion times smaller.
These sorts of complexities are one good reason why circuit theory was developed in the first place. In cases where there is a conductive path, like through a wire and lumped components, connected across a source, circuit theory will allow us to get the answer to all of the questions we need to ask in order to work with electricity. When the frequency gets too high for lumped constants then transmission line theory can be used.
And when both of those fail, give the problem to someone else. 
I was actually referring to the characteristic impedance of the line, but you could more generally consider intrinsic (electromagnetic) impedance fo the medium as well.
Napier’s explaination of what’s going on with electrical transmission is correct, but delves down into QED, which is never a happy subject to attempt to describe in layman’s terms. The last time I tried to explain QED, my date went to the bathroom and never returned. Then there was the time when I started talking about using public key signature to secure electronic voting…yeah, I’m a riot at parties.
Stranger
That was sort of my point. Again, line characteristic impedance is a circuit and not an EM wave derivation.
If you’re going to do it the hard way don’t be a wuss, go hard all the way. 
Line impedance is different than circuit impedance, though. The former is based upon the properties of the line and the amount of power lost due to reactive power loss per unit length. You could idealize it as a virtual resistor and inductor in series with a resistor and capacitor in parallel (see the telegrapher equation) but it’s just properties of the line and surrounding medium. Circuit impedance is based on literal elements in the AC circuit.
Me, I’ll stick to mechanisms you can actually watch strain and break.) I did my time in E&M; not my favorite class. 
Stranger
Your next party is really going to swing.
Yeah, but once you have derived the charactheristic impedance, propagation constant, etc. using EM theory, you treat the line as just another circuit element.
That is, if you know the characteristic impedance and propogation constant you make the computations using those and you don’t start from scratch with EM theory each time you work a problem involving the line.
Even those who can do EM problems in their sleep avoid them whenever possible.