If you're interested, look at the diagram in the "Alternate port functions" section of the ATmega328 datasheet (section 14.3 in my copy). The digital and analogue circuitry is simply connected together at the pin, so the A0, A1 etc pins are permanently digital I/Os at the same time as being analogue inputs.
There's nothing to stop you making the pin a digital-output, and then doing an analogue read (although the result won't be very interesting). Try it. In order to allow the ADC input to float the digital circuitry must be set as an input so that it doesn't "pull" on the pin voltage. Setting pins A0-A5 as digital inputs doesn't stop them being analogue inputs, because they are always analogue inputs.