Why does ice fragment vertically?

Those are some pretty unusual chunks of ice - as you can see, the boundaries of the 'columns' are visible before he breaks them up. So what you've got is a crystalline solid which clearly is the merging of many "source" crystal starting points, or "seeds" as mentioned in the comments.

I can say with certainty from my many years of shinny on frozen ponds that ice does not typically form that way, or at least not at that small scale. I'm going out on a limb (but not on thin ice :-) ) and suggesting that there was a lot of pollen or other small-size debris on your pool which acted as the seed locations for this ice.