Calling it a digital amplifier is a lot closer to the truth than calling an analog amp.Not if you talk to a designer of a class D amp :)
Switching is a bit different from digital. Digital has bits with which meaning (a sound signal) is imposed- how many bits are on or off and in what combination carries the meaning. The switching of a class D amp does not. Instead, it switches according to its encoding scheme. Gain is not represented by how many bits are on or off, gain is instead produced by the encoding scheme, for example in a Pulse Width Modulation scheme the gain is essentially the difference of the triangle wave vs the input signal. Thus it is possible to vary the gain of the amp by changing the amplitude of the triangle wave. This is fundamentally different from digital!