One recent morning in Phoenix, Juan Soto, a star outfielder for the New York Yankees, was watching TV in the visiting clubhouse when an attendant approached him to ask a favor.

In the attendant’s hand were a ballpoint pen and an unblemished baseball, protectively placed in a plastic bag — the telltale ingredients of an autograph request.

The attendant wasn’t asking for himself, however. “It’s for Zac Gallen,” he told Mr. Soto, referring to the star pitcher for the Arizona Diamondbacks. Mr. Soto barely looked away from the screen as he accepted the ball and pen. By the end of the day, a Soto-signed baseball awaited Mr. Gallen in his locker.

The interaction was performed with nonchalance. Among players in Major League Baseball, requesting a fellow player’s autograph is nearly as common as being asked for one by a fan. The players, aware of how fleeting careers can be, collect memorabilia in the truest sense of the term — a way to remember the greats with whom they shared the field.

“It dawned upon me that I’m not going to play forever,” Mr. Gallen said, “and I’m not going to have the opportunity to get guys I’ve competed against.”