Inductor Voltage and Current Phase Delay on a DC Converter
We learn that inductors introduce phase delay between voltage and
  current for AC.


And before that you should have learnt this: -

$$V=L\dfrac{di}{dt}$$

Because that is the fundamental property that governs the AC sinusoidal phase response AND the response in a circuit like a DC to DC converter.


  How would one go about measuring the phase between the two (in the
  context of a Boost converter)?


It's irrelevant because the formula above (from Faraday) tells you directly all you need to know to understand the waveform shapes in a converter. You apply a voltage to an inductor and the current ramps up (or down) at a fixed rate determined by V/L.


  Is there even a phase shift here?


You are asking the wrong question - you should be asking something more like this: are the ramp rates of current corresponding with what V/L tells us.