me or I

The first woman is using the word "than" as a preposition, so its object "me" has the proper case.

The second woman is using the word than as a conjunction to mean "You have way better taste than I (have)." so the subject "I" of the dependent clause has the proper case.

Such sentences don't always mean the same thing:

"Mary likes you more than I" means "Mary likes you mo

So both the sentences are correct then.

The first woman also suggested the sentence "You have way better taste than I do" to her friend. I am assuming that this one is correct too?

There is no potential ambiguity in the case of the original sentence, but a pedant might choose "I" over "me" anyway. Personally, I would add the auxiliary "do" after "I" in the second version. Ending a sentence with a pronoun in the nominative (subject) case, while perfectly grammatical, may seem somewhat pretentious or artificial in today's English.

Anonymous"You have way better taste than I do" to her friend. I am assuming that this one is correct too?

Yes.

If you are taking an English test, especially in a "classical" context, choose "I."
In Victorian English, "me" would be considered incorrect.
In everyday contemporary English conversation, "I" (without the following "do") sounds awfu

Thanks a lot everyone!