I don't think there is any specific feature that makes potatoes unique.

Two chromozome copies are typical for animals, but the number of copies in plants varies widely, and and often changes easily, even different variants of the same crop can have different numbers of copies.

Underground storage organs are nothing unique, and those of potatoes are not even particularly large. Many places prohibit fig trees for example, because what you see is basically just the tip, and there can be a giant 100m in diameter underground that ruins every underground structure in its path. But it's full of nasty toxic sap, and harvesting it would be a nightmare.

It's just the particular combination of fast growth, edibility, and ease of cloning and harvest that makes potatoes unique.