3/8" (0.375", 9.56mm) suggests unified thread, 3/8 UNC = 16 tpi and 3/8 UNF = 20tpi (unified coarse and fine respectively)
'Standard/coarse' M8 threads are 1.25mm pitch which is a gnats over 20TPI
M9 does exist* (lots of fun to find though!) and the 'standard' pitch is also 1.25
1/8" BSP is 28tpi with a major diameter of ~9.7mm. There's also the Septic equivilant... 1/8 NPT is tantalisingly close but has a pitch of 27 tpi
While not impossible it'd be very unusual (and cruel) to make an UNF fastener with a metric 5mm allen key head and 5mm doesn't have a common imperial equivilant...
* there's about eleventy billion metric threads including half sizes like m3.5 and way too many thread pitches- often 3 different fine pitches so M10 for example can have a pitch of 0.75, 1.0, 1.25 or 1.5