Could we use "thank you too" for the response to someone has thanked us? Is it grammatically correct to use that?
It depends on the situation, I guess, but in most cases, saying that would be rather unnatural. We usually say “thank you” when someone has done something for us, and we wish to express gratitude for that action. In other words, it’s usually a one-sided remark. “You’re welcome,” “no problem,” and other variations used as a response to thanks express a sentiment that said action was no big deal.
A: Thank you for cooking this meal for me. (expresses gratitude for person B’s cooking)
B: You’re welcome! It was no problem at all. (expresses that person A is welcome to ask them to cook again)
Thus, saying “thank you too” as a response would not usually make very much sense, because it’s not clear what you’re thanking the person back for. Going back to the example above, it wouldn’t make sense for person B to thank person A for anything, really, because person A didn’t actually do any sort of action.