The FX2 is a 48MHz 8051 (apparently resembles some of Maxim's 8051 chips in timing) with 8KByes worth of FIFO grafted into it. It also has a 16 bit wide fifo interface that can run at 30+MHz.
HID refers to "Human Interface Devices" - protocol designed by the USB-IF committee to handle common computer peripherals. It usually doesn't deal with raw interfaces (i.e. what you'd want if you wanted to just pump data), so a custom driver is usually required. The FTDI and like chips provide a serial port interface, so that's why they use the HID, but as you've noticed, they have a max bandwidth limitation.
Drivers tend to be the ugly thing - I'm very comfortable using Linux, so using the "libusb" style interface, I've been able to dump out ~8MByte/sec data without having to do a kernel-mode driver. On the Windows side of things, I think Cypress has some third party drivers, but I haven't looked in depth at the stuff.
(Incidentally, this is using a relatively recent 2GHz AMD machine, 2KByte "bulk transfer" packets).