Does the gauge of wire make any difference when wiring the inside of an electric guitar

For all practical purposes, there is no difference between stranded and solid wire. The only exception is for very high speed signals (>1 GHz), where solid is arguably better.

The main purpose of stranded is to make the wire more flexible, and has nothing to do with AC resistance at high frequencies.

There is something called the Skin Effect, where high frequency signals tend to travel on the surface of the wire. But stranded wire behaves as a solid conductor in this regard, because the electrons can travel freely from one strand to another. Litz Wire, on the other hand, has insulation between the individual strands and so does behave better with high frequency signals (how much better is open for argument). I have never seen a commercial product use Litz Wire (ignoring audiophile people). I'm sure there is a practical use for Litz wire, but suffice it to say that it is rare.

How does this current carry the data?

Current and voltage are inseperable. The current is flowing because there is a voltage on the wire, and there is a conductive path from that voltage to a lower voltage.

So we can say the data is encoded as voltage pulses or current pulses, it doesn't really matter. Often a high voltage (5 V) indicates a "1" and a low voltage (0 V) indicates a "0". But you could choose any two voltages you like. 3.3 and 0 V. 0 and 3.3 V. -0.8 and -1.2 V. According to what works best in your design.

I read somewhere that wires transmit data almost at the speed of light. How? What is carrying my data? Only EM waves travel this fast.

Another way to look at things is that the voltage at a location on the wire is just a simpler way of looking at the fact that there is an electric field between the wire and everything around it.

When a signal propagates along a wire, it's actually the electromagnetic field between the wire and a nearby "ground" or "return" conductor that is propagating. So it is actually an EM wave, not a massive object (like an electron) that is carrying the signal along the wire.