Actually, meaning determines if it's being used as a conjunction or preposition. If you say "I," you're saying, "He has more strength than I have." (You wouldn't say, "He has more strength than me have.") If you say "me," you're saying, "He has more strength than he has me," which makes little sense. Clearly, the sentence here intends to say, "He has more strength than I have," so "I" would be the correct pronoun, and "than" is a conjunction.
Many years ago, in my freshman high school English class, I was taught that a statement such as "He has more strength than me." was grammatically incorrect. It should be "He has more strength than I have," but "He has more strength than I" was acceptable." (The word "have" was understood from the preceding use of the word "has.") This structure was emphasized for four years and again in freshman English in college. Language evolves. I still prefer "than I," but I haven't tried that in Duo for obvious reasons.
The second have is not translating the sentence literally. For one thing, in Italian there was only one 'ha' in the sentence. You said has/have twice from one word.
Also, one could argue that 'than I' is a relative clause and can take the 'have' verb, but not in Italian. Di is a preposition, and so can not take a verb. In English, 'than' is a preposition AND conjunction, meaning it can take verbs. So when translating, you must keep the original meaning without the have, even though if it were originally in English as what you said, it would be right.
I hope that makes sense. :/