What type of nut do I have here?
You need a 3/8"x26tpi (threads per inch) axle nut. For decades they've all had 15mm wrench flats. (I believe that is something where you can find other examples in very old American, British, and probably other inch country bikes, but that is here nor there.) Bike shops have them.

3/8x24 is usually only for internally geared and coaster brake axles. Mismatching 24 and 26 tpi in either direction tends to go on with force but is likely to damage the axle threads and/or not hold well.

As for how it came to be that one of the most common types of axle nut has imperial threads and metric flats, there are a lot of other examples with bikes of the same thing. I don't know the story in this case and I wonder if anyone alive knows definitively, but factors can include costs for changing tooling at factories, consideration to what tools are in use where. For whatever reason, 15mm has been the nearly universal standard for axle nut wrench flats for a very long time, and that sort of thing has a way of being self-perpetuating because everyone's tooled for it. Meanwhile there's always clearly been some kind of economic draw to the big Asian producers of strictly OEM-level hubs staying with 3/8 axles and not switching to M10x1, a very functionally similar size that many higher quality hubs use. I don't know exactly what that is on the micro level, but on the macro you can bet it's just another example of the bike industry always running under-capitalized and barely in the black if at all. Regardless of what created the initial circumstances, if it's not causing huge problems and it's not going to be a selling point, there's not much impetus to change things just for rationality's sake.