How to identify the type of a microcontroller?
If you're only looking at the network traffic, then no, different microcontrollers don't leave different "fingerprints" on the byte streams somehow.

How a device behaves on the network is a function of its firmware, not hardware.  The hardware has to do the minimum things for the device to work, but the network stack above the MAC layer is going to be implemented in firmware on small and cheap devices.

There are standards for what the various layers in the protocol stack are supposed to do, so mostly different implementations will do the same thing.  Still, there is some wiggle room, and different implementations will have different quirks while adhering to the standards.  You might be able to identify some of these quirks and infer some implementation inside, but that won't be easy and will never be definative.