That biology can be called 'the study of complex things' does not imply that all complex things fall under biology. It's silly to even have to point this out.
No, it's not a formal fallacy. "Apples are red, therefore everything red is an apple" would be. But the word "is" here being used to indicate definition, not characteristic.
Doesn't it? How could it not? Can you give me another example from common parlance of the form "X is the study of Y" where many Y do not fall under X?