Pretty much just a variation of counting sort with a worse name? https://en.wikipedia.org/wiki/Counting_sort
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.
That's talking about the single-pass American flag sort. If you use American flag sort to sort strings, you do multiple passes: on the first pass, you sort all strings by their first character, then for each starting character, you sort the strings with that starting character (which are now in a contiguous subarray) by their second character, and so on.
@chowells calls it a “top-down radix sort” below, which is a great description. It also explains the strengths and weaknesses of the two algorithms: radix sort works great for small strings of fixed length (e.g. IPv4 addresses, which can be thought of as 4-byte sequences) while American flag sort works great for variable-length strings like actual textual strings, especially if they don't share common prefixes (e.g. dictionary words, usernames, etc.)
@hinkley pointed out that the recursive version is just a bucket sort, but in-place. Which is also true!
tl;dr:
Single-pass American flag sort = in-place counting sort
Recursive American flag sort = in-place bucket sort