Normally, you would have "Saya makan + makan siang", but Indonesian don't like to repeat the things that are obvious, so it becomes "Saya makan siang". Makan siang" becomes a verb here meaning, "to have/eat lunch", where English needs the verb + the complement to say the same thing. It's not repeated in Indonesian.
Not in this sentence, I have lunch, and I eat lunch, in this context is the same. "Have" is not = to eat in English, but in this kind of sentence, in this sentence, it is!
I think English speaking people would say rather "I have lunch" that "I eat lunch", that sounds I think unnatural.