"Develop" in this context carries the connotation that it is something that originated inside her body and had no external cause. Anemia is regarded as an illness caused by other factors, so it is "caused within".
The same thinking goes into the expression "to develop symptoms".
In contrast, a cold is something you catch or contract. I think the expression came about when the precise pathology of colds was not well understood, so the old thinking about colds having direct causes probably contributed; certainly, the external symptoms of a cold (which is what everyone thinks about when they think about having a cold) are caused the body's immune reaction, not by the infection itself. But you can still "develop a fever".