Characters in a string can be thought of as digits in a base-256 number. Call counting sort recursively on each bucket, looking at the next character in the string. Can you not see the similarity?
But where the classic radix sort is bottom-up, the American flag sort is top-down. That really matters for data where the the most significant portions of the data is likely to be all you need to consider for the sorting. While a top-down sort will have similar performance to bottom-up in the worst case, in the best case it can skip a lot of passes, especially if the inputs are much longer than the longest common prefix.