Categorizing ICs as digital or analog
It's blurry because it is not black and white, or rather there is a huge grey scale between black and white that sometimes can be considered analog and other times can be considered digital depending on the application.

In reality everything, except perhaps at the atomic level, is actually analog. 

Digital circuits are actually analog comparators, usually with some hysteresis, that makes them flip their outputs depending on what the input voltage or current does. 

Once you have such a "gate" you can combine them in an infinite number of ways to produce more complex logic functions.

Some devices, like simple latches etc are purely logic gates. Others are hybrids that contain both "digital" and analog circuits. 

A simple transistor can be used either as a current amplifier, or it can be used as a switch. Or more accurately, a transistor has three states, Saturated (ON), Reverse Biased (OFF), and Linear (neither on nor off). As such, which mode you are using is dictated by how it is driven and arranged. When you are using it as a switch, using the On and Off states, the transistor always has to goes through that linear region during the transition. So yes, the definition is blurry.

As for your indicated device. It falls into that hybrid category. It takes digital inputs and outputs analog voltages based on those inputs and what the load is doing. 

In the end, as Andy pointed out, it usually doesn't matter what you call it. 

There are some exceptions though. A good example of that is a multiplexer, a device that channels inputs to outputs based on come control signals. A digital multiplexer is very different from an ANALOG multiplexer.