The handing over of an noteworthy home run ball to a player who hit it is apparently a tradition that has existed for some time, as evidenced by the above 1957 Associated Press photograph. (eBay Auction Link) From the August 1st game, the above pic features Brooklyn first baseman Gil Hodges with the fan who caught the record breaking ball. Per the descriptor on the reverse:
GIL HODGES gets ball he hit for record 13th grand slam yesterday from Robert Chilla, 28, who recovered it in left field bleachers. Chilla got new ball in exchange.