I'm fairly certain the crystals are calcium oxalate

....showing crystals of calcium oxalate (black-edged rectangles). These crystals embedded in the cell cytoplasm are known as raphides. Calcium oxalate crystals are common in the stems and leaves of many plants; they have no known metabolic function other than to serve as a means of disposal of excess calcium in the plant. source

Calcium oxalate can form kidney stones!

And, also this...

Garlic can be purple because of two reasons: it is one of the “hardneck” varieties of garlic with purple stripes or because your white garlic reacted with something acidic. Both are safe to eat and can be used in a variety of dishes. source

So I imagine that the purple garlic comes from the variety, and not the crystals