SmartOS is comprised of the Illumos kernel (with ZFS, DTrace, OS-level virtualization and next-generation KVM) with BSD package management and a GNU toolchain.
I used Solaris for almost 5 years and when Sun was bought I thought for sure the cool parts of it were doomed. Illumos popped up and I thought that they didn't have a chance, it would wither and die. It's very nice to see that is not the case.
Considering that the Illumos kernel is basically the OpenSolaris kernel, I don't understand what you mean. They are using Solaris, but for some reason they're trying to hide it.
I suppose they could refer to it in technical documents as "the OpenSolaris kernel", but by that same logic, an OS using the DragonFly BSD kernel should say it's using "the FreeBSD kernel".