What properties should I look for in a microcontroller if I just want to work with incoming 16 bit data?
An 8 bit micro can do 128 bit complex floating point matrix math, just very slowly. Any CPU can perform any computation given enough time and space, due to the Church Turing thesis.

RAM and serial communication speed is often more of a constraint than processing, but that can change if you need to do lots of digital filtering. If it's just moving 4000 samples per second on to a serial port, then nearly any uC around 8+ MHz will be able to spend 2000 cycles per sample, which is plenty. Given the low cost of 32 bit ARM uCs these days, often with Bluetooth and analog inputs, you might even find a lower cost solution with a more powerful CPU.