It depends whether you're asking about English, or about the artificial language made up by grammarians a couple of hundred years ago on the basis of "If only this were Latin", and inflicted on schoolchildren ever since.
In the real English that most people speak,
My dog can run much faster than me.
is normal and grammatical, and the "than I" version sounds stilted and unnatural. The existence of "than I can" is irrelevant, as the 'than' has a different grammatical role.
In the made-up English that the test probably expects, "than I" is the only correct answer. That shows you how made-up it is.