Can an optical mouse sensor chip (System On Chip) which has a digital signal processor, Serial I/O, and a set of registers be considered as a Micro-controller?
IMHO As long as they have a CPU, memory, peripheral units and are programmable can be considered as micro-controller. But as captcha mentioned today most of them are a not programmable or so little programmable-configurable that only let companies to vary their products with little tricks.
It probably started out as one in the development stage but I guess that nowadays these chips are purpose-built chips that are far less flexible than the average micro is. To answer your question.. No.
Yes and no. There are many dedicated ICs which are not specially built parts, but really generic controllers with specific programming. Anything with a 8051 core is one example. Or a more specific example is MicroChips mcp2000 (p/n?) Usb to serial IC. Its well known to be a PIC18fxxxx, a usb capable microcontroller with Microchips USB to Serial firmware, but licensed for general use. It even registers as a pic when hooked up to a pic programmer.
But would you call it a microcontroller? Technically, it is. In practice and marketing, it is not. You would not reflash it with custom software, and will always cost more than a blank mcu.