I think that Brother would be spring. I think this because spring is a season that nourishes and replenishes the land. It brings an awakening to nature after its dormant winter slumber. It causes things to bloom and grow. Spring is a season that brings renewal. However, spring can also be devastating. The thawing snows and spring rains can cause rapid flooding. Spring storms can be quite violent and loaded with lots of lightning.
Brother would be spring, because he is that mixture of wonderful growth and danger. For most of the story, Brother works with Doodle to improve Doodle's quality of life. He teaches him to walk, run, climb trees, etc. Nobody thought Doodle would be able to do those things, but Brother's work and dedication caused Doodle to grow in ways that nobody expected.
Unfortunately, Brother wound up being deadly to Doodle as well. During one of their brotherly forays, a huge lightning storm came upon them quickly. The boys began running home, but Doodle could not keep up. Instead of staying with Doodle, Brother ran harder and faster.
"Brother, Brother, don't leave me! Don't leave me!"
The knowledge that our plans had come to nothing was bitter, and that streak of cruelty within me awakened. I ran as fast as I could, leaving him far behind with a wall of rain dividing us. Soon I could hear his voice no more."
Doodle never caught up. Brother worked his way back and discovered that Doodle had been killed by the storm. Brother had worked to grow Doodle, but failed to protect him. That's how spring is. It replenishes the land, but can also cause devastation.