I think the OP is somewhat confused because he is viewing this from the wrong angle. He's using controller vs (or) processor as some basic level of complexity distinction. Eg, an application processor would be an ASIC that has some CPU, external memory, MMU with VM, page table, DMA, etc... A microcontroller on the other hand would have more simple peripherals and lack things like an MMU or DMA controller. (simplification). Basically, that a controller has features suited to IO, while a processor has features suited to computational performance.
FPGA's, being reconfigurable, can fill a lot of these. Some FPGAs have hard-ip ARM cores, with FPU, MMU, and DMA features as well as a lot of simple IO functions (i2c, spi, rs232, ethernet, etc...) These are intended to be used as processing systems with the FPGA as custom HW accelerators. They might not always be used that way, but that is the intent. FPGAs can also implement a variety of processor/controllers.
However, the third option is "neither". FPGAs can fill this as well. In this case, the FPGA would only be HW accelerators engines with basic (logic gates, not cpu based) control between modules. This is very high performance for simple or specific applications.