The unit itself doesn't actually matter. Even industries with the least precision set their stuff up with so much precision that the unit you use basically doesn't matter.
Your machine may spit out widgets that are plus or minus an inch. But when you set up the machine you set it up to the 1/16 regardless. Swapping all that to metric doesn't actually change anything other than the number the guy setting it up dials it in to.
1/16” is just over 1.5 mm, so yes, the guy setting the machine in millimeters is giving you more precision. In the real world measurements aren’t just abstract figures you can move around losslessly.
I have a socket set in half-millimeter sizes for the absolute plague of cheap bolts and nuts that are being manufactured with obscene levels of slop.