Does this lyrics mean something like never leave you alone?
Native speaker’s answer
Rebecca
Yes, it is! "On one's own" means to be unaccompanied by others, so it refers to the status of someone being alone. The synonyms are "alone", "all alone", and "by oneself." Ex: He went to the game on his own. Ex: He went to the game by himself. Ex: He went to the game alone.