Thread sizes below 1/4 confusing

[Mattax]

That is correct. Per ASME, or per SAE, there is no such thing. It's a #10.

Considering the location,
the question is whether he got 3/16-24 BSW with its 55* thread form,
or did he get a 10-24 UNC with our 60* thread form.

Will they physically thread together? Maybe, depending on the running fit.
Will they torque properly? Probably not.