How to order a list to match the order of another list?

Using Ordering it's possible to do this. Ordering can be seen as a permutation that brings a list to the identity permutation. And applying Ordering again to this permutation we get the inverse permutation that brings a list from identity to the original order. Using these observations:

OrderingToTarget[list_, sourceIds_, targetIds_] :=
  list[[Ordering @ sourceIds]][[Ordering @ Ordering@targetIds]]

OrderingToTarget[x, x, y]

Inspired by Sort two lists at the same time, based on another here is a different approach:

permuteLike[a_, b_] :=
  Module[{x = a},
    x[[Ordering @ b]] = Sort[a];
    x
  ]

This proves to be somewhat faster and I find it somehow more pleasing as well.

a = {"f", "a", "h", "y", "c", "d"};
b = {"w", "t", "q", "o", "b", "p"};

OrderingToTarget[a, a, b]

a ~permuteLike~ b

{"y", "h", "f", "c", "a", "d"}

{"y", "h", "f", "c", "a", "d"}

{a, b} = RandomInteger[3*^6, {2, 1*^6}];

OrderingToTarget[a, a, b] // RepeatedTiming // First

a ~permuteLike~ b         // RepeatedTiming // First

0.3234

0.228