There is not a word specifically meaning "how manyth" that is in common usage in America. The closest word, though it would be ambiguous, is which, as in Barack Obama is which American President? But you might get an answer like "The current one" or "The black one" or "The one who did the big health care law" and those would all be considered reasonable answers.
If you make it clear you're asking for the ordinal number by giving context, then "which one" is a common usage: Obama is the 44th POTUS. Which one was Hoover? It gets clearer if you ask which number president is Obama? but I do not like how that sounds because people generally take "number" to mean a cardinal number (like 44) not an ordinal (like 44th). While you could ask which ordinal number president is Obama? only a mathematician would consider that a reasonable phrasing and most other people wouldn't even understand it.
Because we do not have a word specifically for it, phrasing the question to get the answer you want in a way that is unambiguous and strictly grammatically correct gets does get somewhat convoluted and sounds awkward, but since we all know there isn't a better way to ask, we accept it.
Personally, when speaking informally, I'd ask like this:
Me: Which president is Obama?
He: The current one.
Me: I mean which number? Like the 40th or something?
When writing formally, I'd work around it with something like
G. H. W. Bush was the 41st president, G. W. Bush was the 43rd president, but which one was J. F. Kennedy?