"Come back" is a phrasal verb that means to return. It is very common and also informal. It is not old-fashioned and you can say it to anyone. https://www.macmillandictionary.com/us/dictionary/american/come-back "Come back" is used in the imperative form in your example sentence. "See you later" is another phrase we use when saying goodbye, but it is a statement, not a command, so it does not mean the same thing as "come back."

Come back soon is something you would say to a customer when they are leaving. But I think it would be a little old-fashioned these days. See you later is more informal and more common.