This sounds like a crude definition of a microcontroller. It has GPIOs and can fetch instructions from memory. Hardware requirements are based on the following:
memory: How much playback do you need? The Salae uses 10bit samples at up to 24MHz, or 30MB/s (also the single channel USB 2.0 limit). You may require external memory modules to support long traces. This can be done directly from USB, with minimal on-board memory, and would depend on your computer's USB bandwidth, like the Salae does. If you're using both at once your computer's USB controller may reach its limit. I wouldn't want to depend on this.
IO speed: Producing multiple 10-bit 24MHz patterns is no small feat. Look for chips with many hardware PWM modules. You may need to use multiple slave PWM chips. How much greater the clock speed must be than the PWM frequency depends on the chip family.
noise: On a 5V digital system, an 8-bit LSB is about 20mV and a 10-bit LSB is about 5mV. For these bits to be effective your noise level must be below that. This will be important when running so many parallel conductors, all with 24MHz digital* signals. Although this speed is not high, it can have >20mV crosstalk if traces are coupled. I don't think special drivers (bipolar, etc.) or EMI protection is required - just space them out.
USB: Look for hardware USB support, or go with an external FDTI chip. It can be done in software as well, and is an option at least for M3s and XMOS, but keep in mind it reduces thread slices for your PWM memory fetching, etc., especially if it is being used to fetch PWM data from USB (limited on-board memory).
Possibilities include the PIC24 family, XMOS, and many of the ARM family. There are some great parametric tables out there: RS, Microchip. It would be worth getting to know the differences between ARM implementations to see if one will do everything.
Note that you may also want to drop to 8-bit PWM, as you could then lower your hardware requirements.
*Yes, they are analog, but they're also a recording of a digital signal. Hopefully they appear so.