"Take On Me" - Scandinavianism or valid English?

It's a music lyric and hence has some liberty.

But "take on" is a fairly common idiom in several contexts. It can mean "engage in combat", but a less aggressive meaning is to accept some burden or responsibility. Google finds "she didn't want to take on more responsibilities", "why would I want to take on an apprentice", "why did I want to take on the role as a GM", "I want to take on Usain Bolt's legacy", "does someone want to take on this project".

"Take on me" can thus be considered to be a plea for "you" to take on the burdens and responsibilities of having a relationship with "me". Or it can simply be considered a poetic rewording of "take me on" (which can itself be interpreted in a number of ways).