"I am me," and "it's me" are both colloquial but are not proper English. The verb "to be" is a copulative verb (linking verb) so unlike transitive and intransitive verbs, there's neither a direct object nor an indirect object.
The correct statements would be "It is I" and "I am I," but the first one sounds pretentious these days, and I'd likely say "I am" instead of "it's me" so as not to sound like a 19th century play or someone who uses bad grammar.
Sorry mate, but "I am I" is more than accepted. I say, "It's me" like everyone else but that is entirely diferent from "I am I. " The former is simple identification the latter is a declaration of autonomy. "I am I; others are others," is a sentence in which I would not use "me."
Tôi = 1S pronoun (I/me)
Là = am/is/are (Vietnamese doesn't conjugate for person)
As for translating from Vietnamese, use whichever is correct in English. "Me is I" or something like that as a translation of "Tôi là tôi" would make no sense whatsoever, so it would obviously be "I am I" or "I am me".