Should I refer to North Koreans and South Koreans "same people" or "the same people"?

You would use the definite article there:

North Koreans and South Koreans are the same people.

Edit

Same without the article is often used in a telegraphic style (one that omits extra words for the sake of brevity), such as product orders:

"You requested 52 greeting cards: will ship same within 4 days."

In this case it functions as a demonstrative pronoun, whose antecedent is the cards mentioned. Another way to say that is

"You requested 52 greeting cards: we will ship those to you within 4 days."

It fine to use either, but same in that context is a conventional usage.