Use of than

We hear in daily conversation: I like it more than him. I like it more than he.  Another example:  Art likes Mary more than I.  Art likes Mary more than me.  Meanings of each sentence define the use of  I or me.

A preposition combines with a noun or pronoun to form a phrase that modifies an object or action: by herself, before sunset, behind the door.

The noun or pronoun becomes the "object of the preposition." So in the example behind the door, the word door is the object of the preposition behind. And when you're using a pronoun as the object of the preposition, it must be in the object case. In other words, you use object pronouns such as him, her, and us. Behind her; behind her; behind us.

Some grammarians argue that than is a conjunction. They say the case of the pronoun after the word "than" is determined by its role. Conjunctionists argue that the sentences: Art likes Mary more than I and Art likes Mary more than me are both correct but have entirely different meanings.

Both use than as a conjunction, but when you use the subject pronoun I, you're saying I like Art more than I [ like Mary], and when you use the object pronoun me, you're saying Art likes Mary more than [Art likes me].

If than is a preposition, always use the objective pronoun me and  the same sentence would mean both things--you don't care for Mary as much as Art does AND Art prefers Mary to you.
If you wish to avoid ambiguity, clarify the sentence by finishing it with the verb added. Mary is wiser than I am. He likes her more than he likes me. With a verb present, the choice is obvious.