380 and 9mm are actually .355, but I know what you mean.An economical vernier caliper capable of making accurate measurements to 0.001 of an inch [0.0254 mm] wasn't invented until 1851 when it was introduced by Joseph R. Brown [Brown and Sharpe] of Providence, RI. It was not until 1867 that Brown perfected the micrometer caliper. These are two of the most basic modern machinist’s measuring tools. So, you are correct that simply measuring things prior to and during the Civil War was a problem. But, you can't have weapons in which your bore is smaller than the bullets in your ammunition. The ordnance establishments in armies around the world issued their inspectors "go/no go" gauges to ensure that bores were neither smaller than their established ordnance standard nor larger. Anything that didn't meet standard was supposed to be rejected and condemned. Similarly, gauges for bullets and finished cartridges [where the bullet was loaded in the greased paper wrapper] were issued to the ammunition laboratories.
Re Burnside caliber, caliber designations are sometimes terms of art: the .38 Special, .357 magnum, 9mm, and .380 calibers all essentially measure .357 inches. What matters is true bore diameter and sizing the bullets to that.