If we are to believe York at his word it could be that he picked up a 1903.  However men during that time were issued a gun and the serial number was recorded just like it was shown in the film.  This was their gun and they had better well guarded it with their paychecks or else.  It is difficult to believe one or a few private men would be allowed to have a rifle not of the same model as the rest of the division.

Picking up a unissued rifle would elicit the wrong type of questions from the Officers and NCOs I am certain.

From the History of the 328th.

Specialty schools were established by the Division in charge of
British and French instructors fresh from the fields of battle. Representatives from
each company were trained in these schools and soon each company had men capable
of instructing in auto rifle, hand grenade, rifle grenade, bayonet, gas defense and
sniping. Clothing and equipment was being received and issued daily and soon the
percentage of straw hats and tennis shoes at dress parades was on the decline. Saw
mills of the camp were rapidly turning out the "new model" Camp Gordon rifle and
soon each man was busily occupied in mastering the intricacies of the manual of
arms with this bed slat variety of fire arm. Work continued with this makeshift
equipment till about the first of February, 1918. Then we were fully equipped with
the United States Model 1917 Rifle and Bayonet.

pg 21 ->