the 8051 is not really any more unique than any other modern CPU core, its quite dated actually.
However, it is used royalty free in many SOC designs, and as Hamster said, it is often used in an FPGA design to provide a core alongside the design. Todays larger FPGAs have plenty of room and can save board space if you have the FPGA space.
And when you program it, you'll find that you're going to do most of your work in a high level language anyways, so the cpu core for the most part doesn't matter anyways, except for the little bit of assembler that you might need in an ISR.
Bit twiddling in C using C constructs is almost the same for every micro out there.. just the types of peripherals are different, behave different, have different I/O locations, and different control bits... Thats the stuff that needs to be ported or abstracted away using a driver layer, for the most part.
You should learn something current. If you want to breadboard it up, than start with PIC or AVR as many have suggested, you won't go wrong. There are lots of modern hobbyist friendly PICs and AVR devices. For ARM devices, you can get plenty of small ARM development boards for a very low cost.
As for 8051, just use one of the many software emulators out there to play around and learn the assembler, if that's your goal, but don't put too much value on it.