The 6.3 mm is a rounded-off value in metric system for 1/4-inch in the English system. The conversion is 25.4 mm per inch. That would be 6.35 mm plug.

It's possible that the value is intentionally 6.3 mm instead of 6.35mm to provide some clearance for insertion into the jack. My AKG K140 (old!) headphones has the 1/4-in jack standard, and it measures 6.3 mm plug diameter.

The AKG has BOTH sizes.  there's an included adapter so you can use either jack size.