The reason faulty generalization is an informal fallacy is contained in the very link you provide. The fallacy lies in insufficient empirical evidence, but the same form of argument could become convincing and cogent with enough empirical evidence. For example, suppose someone argues, "I've taken two cab rides and both of the times the driver was rude. Therefore, most cab drivers are rude." The reasoner has gone wrong because two instances is not enough to draw a general conclusion about most cab drivers. But if you gathered enough empirical evidence you could make the argument convincing. By contrast, formal fallacies are invalid and could never become valid no matter what empirical evidence exists.
The formalization you have provided is in fact deductively invalid. But that is not the form of faulty generalization. Faulty generalization has the form of an inductive argument which gathers a number of instances and draws a general conclusion. Consider again the example, "I've taken two cab rides and on both the driver was rude, therefore most cab drivers are rude." This is a faulty generalization, but it doesn't have the form you have given, since it is a conclusion about "most" cab drivers and not "all." Where the reasoning has gone wrong has to do with the fact that the reasoner does not have enough empirical instances to draw her conclusion, but if she had enough instances she could be warranted in drawing the conclusion.
By contrast, a formal fallacy consists in a form of argument that could never be sound or cogent no matter the empirical evidence. For example, if someone says, "All birds are animals. This is an animal. Therefore, it is a bird." No abundance of empirical evidence could make this a sound argument because it is formally invalid.