There is an obvious comparison between the scarlet ibis and Doodle. The scarlet ibis is an exotic bird. The father concludes that a storm must have brought the ibis from the south to their area. The bird looks, and is, clearly out of place. Perhaps because of its journey or its strange surroundings, the bird dies. Likewise, Doodle does not fit in with his surroundings. His family loves him and they do not exclude him in any way but he is different and this makes it more difficult for him to function in this environment. His brother makes it his mission, more out of feelings of embarrassment and selfishness, to make his brother "more normal."
The scarlet ibis failed to adjust to its strange environment. The bird's death is a moment of foreshadowing; one of which none of the characters are aware. Whenever the characters or the narrator are unaware of something but the spectator or reader is aware of it, this is called dramatic irony. The reader recognizes the connection between the ibis and Doodle; Brother does not. The death of both may not be immediately recognized but upon reflection, the connection is clear. The dramatic irony is that Brother doesn't recognize the lesson he might learn from the ibis (what we see as a foreshadowing lesson) and Brother continues to push Doodle to fit in.
We don't know exactly how Doodle dies (maybe the nightshade, maybe the exertion of training). But both Doodle and the bird were forced to adapt. And it is this forced assimilation, (one from a storm, one out of maliciousness), that leads to their deaths.