The phrase "to take somebody on" can also mean to hire someone. E.g. "The company took me on as an apprentice when I finished high school."

Take me on: Usually used in phrases like “Take me on a date” or if someone is competing with another person, you can say “Take me on” meaning compete with me. Take on me: I never really hear this phrase. But you can ask someone “What is your take on me?” And you’d be asking them what they think of you