As far as learning C, not really any advantage. Learning to program in C, you'll be writing command line programs that should run with little or no trouble under Linux or DOS. By avoiding a few libraries and functions, there would be no trouble switching back and forth.
If you can deal with a 16 bit stub (DOSbox fixes this), DJGPP is the GNU/Linux C compiler and much more ported for DOS use. So in other words, you can get virtually all of the exact same tools found under Linux that will run under Windows/DOS. In fact, DJGPP is one way you can compile some Linux programs and have them run under WIndows/DOS. If you do get DJGPP, I also suggest getting RHIDE which is similar to the Borland IDE that was available at the time. Keep in mind that this is all free just like Linux.
If you are developing a Windows program, then obviously Windows is the platform. Similarly, If you are developing a Linux program, you should develop under Linux. The operating system determines where you want to develop usually.
However, if you want to develop something that will compile and run under both or many operating systems, you'll need to stick with the C standard and with minimal differences it should compile and run. If command line programs aren't enough, then you'll need to use one of the cross compatible libraries that are available. There are tons of programs that will compile for a variety of operating systems because someone took the time to port them and fix anything that was incompatible. C has a mechanism to selectively compile parts of the source code.
There is also the option of using an emulator such as WINE under Linux to run some Windows programs. It isn't the best solution but it works for many programs.
So, is there really an advantage? There isn't an answer. There are C programs that will compile under over a dozen variants of different operating systems. To manage this, each program is dynamically modified so that it actually compiles for the specific operating system. Which means that someone had to test and fix the problems for every operating system using the target operating system and the tools available.
Shadow Wolf