KOBE, Japan—When Ichiro Suzuki arrived at Miami Marlins training camp last month, he was the only player on the 40-man roster born in the 1970s. In fact, he was only two years younger than the team’s manager, Mike Redmond.

At 41, Ichiro was trying to hook on with his third major-league team, having parted ways with the Yankees after playing parts of three seasons in the Bronx. But he is no charity case. Indeed, when one of his new teammates, infielder Mike Morse, saw that Ichiro had signed with the Marlins, he sent a congratulatory...