What shift register ICs am I looking for?
The item on the left isn't a shift register at all; it's known as an "addressable latch". 74xx259 is one example.

However, I'm not aware of any off-the-shelf components that have an internal counter; they all expect you to supply the 3-bit binary address for the latch that is to be updated.

It would be easy to create such a device using a PAL or CPLD.

Also, if you have a SPI or I2C bus on your controller (or are willing to bit-bang one on GPIO), there are many kinds of I/O expander chips available for these buses. One example that I've used is the Microchip MCP23S17 (SPI) / MCP23017 (I2C). Each chip gives you 16 more lines of GPIO.