And they were only on M1903 Springfields and (very late) on Krags. Not on M1903A3's or M1917 Enfields or M1 rifles.
To backtrack a bit. A star gauge is an instrument for measuring inside diameter of a tube/barrel. It is essentially a long hollow rod with two pads on one end that are forced outward by a pointed internal rod with a caliper handle (like a micrometer) on the other end. The pads are narrow enough that they can measure either lands or grooves
In use, the gauge is inserted into the muzzle end of the barrel and readings taken every inch. The readings must be within the specifications for those barrels or the barrel is rejected.
In the slow days of peace, apparently every barrel was gauged before being installed. If one proved especially correct and even all the way through, it was stamped with a number on top, a star-gauge mark was made on the muzzle at the 6 o'clock position, and a star-gauge card filled out with the measurements. It was then set aside for use with match rifles.
In wartime, no one had time for such nonsense, and it was eliminated altogether in WWII, not only because it was time consuming but because air gauges were developed for routine inspection. Also, production methods had improved so much that almost all barrels were as good as the best from the star-gauge era.
Jim