This is very likely just pure logic, the bulk of which being counters and registers and yes, a FSM. After that you have LCD drivers, a driver for the beeper, a driver for the backlight LED, debouncing input blocks for the buttons and the oscillator. As the picture below suggests, there may also be a charge pump, probably to generate a bias voltage for the LCD.
As can be seen on the die, this is an OKI IC. Casio started to use OKI watch ICs in the 70's and while I don't know what's current in the latest CASIO watches, I suspect that they still largely use OKI ICs, at least for their simpler models.
Interestingly, you can see a (c) 2006 marking on the die itself, suggesting that this die was a respin of the original IC they used in the late 80's. This is still on a relatively old process node. Likely > 1 µm. The die is also marked "05MG137-A01", which is probably its OKI reference. Couldn't find anything much from this reference, but not too surprising as theses ICs are not "off-the-shelf" stuff.
Being powered from a CR2016, if we take the bottom of the typical capacity (about 80 mAh), 7 years of operation equate to approx. 1.2 µA of average current for the whole watch, probably including occasional use of the backlight and knowing that the LCD is operating 24/7, this wasn't bad at all in the late 80's. It's still not bad by today's standards.