Edit: though I thought about it a little more. The more interesting question is regarding the standard library, rather than the compiler. Applying different licenses to the two is not unusual (gcc is GPL, glibc is LGPL). From what I can see, the standard library is embedded entirely into the compiler, and thus yeah, I do think programs that use this would actually have to be GPL. Not that that's a bad thing though :)
More important than the standard library (which isn‘t so standard, just conventional, see musl) is the GCC Runtime Library, which is GPLv3, but with the "GCC Runtime Library Exception".
There is also an interesting question about license compatibility between GPLv3 and GPLv3 with RLE, but that is mostly ignored by everybody.
Edit: though I thought about it a little more. The more interesting question is regarding the standard library, rather than the compiler. Applying different licenses to the two is not unusual (gcc is GPL, glibc is LGPL). From what I can see, the standard library is embedded entirely into the compiler, and thus yeah, I do think programs that use this would actually have to be GPL. Not that that's a bad thing though :)