Doodle's older brother goes unnamed throughout the story. He is simply known as "Brother." Brother is the story's narrator, and he never introduces himself as anything more than Doodle's older brother. Perhaps it is to keep his anonymity from readers. I think it has more to do with how brother sees himself in the telling of this short story. Brother sees himself as exactly that. He is Doodle's brother and protector. At the beginning of the story, Brother is embarrassed by having a brother that can't walk by age five.
"embarrassed at having a brother that age who couldn't walk"
What's awesome about Brother is that he works to make a change for Doodle. Brother works hard for and with Doodle to help him learn to walk and later run and climb trees. Brother transforms himself into a great brother. He's the brother that many people wish for. Unfortunately, Brother makes one fatal mistake, which likely costs Doodle his life. The guilt that Brother felt must have been terrible, which is perhaps why he never reveals his real name to readers.
The narrator has chosen to leave his name unmentioned. So, we as readers refer to him simply as "Brother".