"'In the American League it is the Yankees as I said,' the old man said happily. 'They lost today,' the boy told him."
The original Yankee Stadium in the year it opened, 1923
Public DomainThe original Yankee Stadium in the year it opened, 1923

The American League and the National League are the "two leagues that make up Major League Baseball in the United States and Canada."

Since we know the story takes place in September and the year is probably 1951 (the book was published in 1952, and 1951 was DiMaggio's last season), below is the Yankees' schedule for September 1951. (Their entire 1951 schedule can be found here.)

We might even attempt to determine the exact day of the story, based on Manolin and Santiago's talk about teams on page 17 and the fact that the Yankees "lost today."

Game

Date / Box Score

Opponent

Score

Decision

Record

128

09-01-1951

at Washington Senators

4-0

W

81-47

129-I

09-03-1951

at Philadelphia Athletics

3-1

W

82-47

130-II

09-03-1951

at Philadelphia Athletics

2-3

L

82-48

131

09-05-1951

vs Boston Red Sox

2-4

L

82-49

132

09-07-1951

vs Washington Senators

4-2

W

83-49

133

09-08-1951

vs Washington Senators

4-0

W

84-49

134-I

09-09-1951

vs Washington Senators

7-5

W

85-49

135-II

09-09-1951

vs Washington Senators

2-0

W

86-49

136-I

09-11-1951

vs St. Louis Browns

3-4

L

86-50

137-II

09-11-1951

vs St. Louis Browns

3-6

L

86-51

138

09-13-1951

vs Detroit Tigers

2-9

L

86-52

139

09-14-1951

vs Detroit Tigers

5-2

W

87-52

140

09-15-1951

vs Detroit Tigers

4-7

L

87-53

141

09-16-1951

vs Cleveland Indians

5-1

W

88-53

142

09-17-1951

vs Cleveland Indians

2-1

W

89-53

143

09-18-1951

vs Chicago White Sox

1-7

L

89-54

144

09-19-1951

vs Chicago White Sox

5-3

W

90-54

145

09-20-1951

vs Chicago White Sox

5-4

W

91-54

146

09-21-1951

at Boston Red Sox

5-1

W

92-54

147

09-22-1951

at Boston Red Sox

0-5

L

92-55

148

09-23-1951

at Boston Red Sox

6-1

W

93-55

149

09-26-1951

vs Philadelphia Athletics

1-4

L

93-56

150-I

09-28-1951

vs Boston Red Sox

8-0

W

94-56

151-II

09-28-1951

vs Boston Red Sox

11-3

W

95-56

152-I

09-29-1951

vs Boston Red Sox

4-0

W

96-56

153-II

09-29-1951

vs Boston Red Sox

3-1

W

97-56

154

09-30-1951

vs Boston Red Sox

3-0

W

98-56