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The CPU in the PlayStation 2 is essentially a MIPS-IV missing a few instructions and with a few proprietary extensions which Sony put in, and it's got some thing which apparently helps to decode MPEG streams.
But one thing I could not find is an etymology! So why call it an EmotionEngine? Emotion is not really a prominent word in Sony branding as far as I can tell. So there must be some kind of history behind this term.
There's a pretty nifty article linked from the Wikipedia article on the Emotion Engine that discusses the details in the "Emotion Is the Difference" section on page 2:
But much of the EE’s compute power will go toward an even loftier goal: behavioral synthesis, or, as SCE calls it, emotion synthesis. This technology gives game programmers the ability to accurately model all manner of physical systems, allowing realistic behavior of characters and objects. For example, the system will enable lifelike facial expressions
There's another description in this paper:
Emotion synthesis means the real-time synthesis of a computer graphics animation scene that projects a great deal of atmosphere. For example, when a female character walks into a video game scene, her motion must be determined by solving physical equations in response to interactive events instead of replaying prerecorded data. Moreover, differential equations with a large number of variables must be used to describe, for example, the waving motions of her hair in a breeze. For authenticity in emotion synthesis, the CPU must execute these calculations in real time.
As far as I know there is no actual behavioral synthesis or emotion synthesis technology used in any PS2 games. It's a generic way of saying "We have a lot of computing power so games will be more realistic". But for marketing purposes it makes a snazzy name that implies there's some special functionality to the EE that competitors don't have.
The CPU core is only one of eight parts of the chip. It has two vector units, an image processor, DMA controller and various other components. Together they form the Emotion Engine, which I recall at the time being marketed as bringing procedural generation technology to games to make them more realistic and varied.
