Oxford Online Dictionary classifies "so" as an "adverb". However, if you read the linked ELU question, So do I ( what word type is “so” functioning as?, you will notice that there are different opinions.
No matter what it is called, its primary function is not to repeat what is said before the word.
A: I love you.
B: I love you, too.
You can say "So do I" in place of "I love you, too" as "(do) so" refers to "love".
You example sentence could be rephrased to:
Just as the nation-state is a modern phenomenon, the explosive increase of the human population is a modern phenomenon, too.
You don't need to repeat "a modern phenomenon" as it is repeated. "So" could replace it as long as it is inverted and placed at the beginning of the clause. The subject of the second clause is not a modern phenomenon.
You can read my answer to this question, the usage of 'so' in ' it does so very slowly'' if you are interested in learning how "so" works in different ways.
