Would adding the redundant hai at the end of the sentence change the meaning of the sentence in anyway? Or is it used just for clarity and completeness to make it a complete sentence instead of a declarative verbal sentence fragment? Also as a follow up, if aaj nahi ja rahii is the object of the sentence, is nahi ja rahii also the verb, or is the verb hai, which is implied in this phrase but not explicitly stated?
It is similar to the 'Royal we' in English.
In the past, certain people of a 'higher social status' used to refer to themselves using हम, especially when speaking formally, and this was considered a sign of being 'well-bred'. This is rarely done in modern standard Hindi, except by some older people. The actor Amitabh Bachchan is one prominent person I can think of who uses हम this way.
In addition, there are some Hindi dialects where हम is the default first-person pronoun for both 'I' and 'we'. मैं is not used in these dialects.