I would rather say "This is something we have in common". In common already implies a connection between the two of you, you don't need to add both.

This is something we both have in common

Badminton

Although John is my teacher, he likes to play badminton. I don't understand this sentence. What is the connection between somebody being my teacher and whether or not he plays badminton? If it were "he likes to play badminton with me" that would make more sense...