There can be a problem when coiling up coax, but it's a mechanical problem. As in winding it too tightly and causing distortion. If the dimensions on the inside of that cable change, so does the impedance of that cable. If that distortion is severe enough, the center conductor will 'migrate' to the side and eventually short to the braid. Neither of those two things are something that's 'good' for your radio.
How tightly you can coil coax cable depends on the cable. There are several different 'sizes' of coax, some are 'stiffer' than others. That 12 inch coil thingy is only a rough 'average' size of the coiled up cable, that can certainly be too tight for some cables, and very 'loose' for others. For the usual type of coax that people normally use, it's a fairly nice average size of coil. If you really have to force a cable into a particular sized coil, then it's probably too tight.
With coaxial cable, the inside is very different from the outside, electrically. Like a pipe. You can run a pipe of pure water through really nasty water and the stuff inside that pipe stays pure, right? Same thing with electrical signals and coax cable. Also like a pipe, things can travel on the outside of it and get into what that pipe carries and contaminate it where there's an 'opening', or at the ends of the pipe. The 'trick' is to not furnish the electrical thingys on the outside of that coax an opening to get into what that coax is carrying, keep it 'sealed'. Sealing things electrically is very different than sealing things mechanically.
RF signals are alternating current, AC, not direct current, DC. So there are some electrical characteristics that play a part in that "keeping the insides of that pipe" clean. Inductance, coils, are one way of doing that. A coil, or inductance, impedes, or slows, or stops, AC. So, if you wind a coax cable into a coil you produce some inductance on the "OUTSIDE" of that coax. That inductance slows or stops electrical signals from flowing on the 'outside' of that cable. Which is good, and the whole idea of an RF 'choke'! It chokes off that stuff on the outside of the cable. It has no affect on what's inside that cable unless that coil distorts the cable which changes it's impedance which means you wound the @#$ thing too tight (you can bend a pipe too much and it splits, right?).
That's a very 'rough' explanation, not very scientific or exact, so take it as such. To complicate things jsut a bit, the outside of coax cable can be used to conduct things/signals that you do not want to mix with what's inside that cable and you have to keep those two kinds of signals separated. It's done quite often. The 'trick' with that is to keep them separated by using other electrical characteristics.

- 'Doc