It’s what’s called a M3 (3mm.) Hex nut. Which is 5.3 mm measured across the flats.
Based on your photo of the brass nut it measures 5.33 across the flats. A 5.5 mm socket gives you .17mm. over size clearance . It’s called a M3 because it fits a 3 mm bolt thread. Hex nuts are sized by the diameter of the threaded bolt they’re going on. When choosing a wrench you look for a M3 socket which measures 5.5 mm opening across the flats.
It’s that simple. Trust me. Oh yeah if it’s a Chinese product it’s a metric thread. I’m a retired plumber, steamfitter. I’ve likely installed a million nuts and bolts of various sizes.
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