It is a name. The identity comes from the rituals, beliefs, clothing, literature, temples, architecture, food and so many other cultural and civilizational markers.

Pakistanis who were looking for their fathers in Arabia, Persia and Turkey in the past based upon a deliberate avoidance of their Indic origin have recently started a political project to reclaim their identity based upon the IVC. They can try all they want but civilizational continuity is the only thing that matters. Names are dime a dozen. If you have a problem with Hindu/India and other exonyms, you could always call yourself Bhāratīya or some such. It is all the same.

Honestly not really. Hindu was the widely used demonym to refer to any Indian before Hindu nationalists decided to turn it into a religious designation. To be a follower of the Hindu religion meant that you followed some set of practices that originated in India. There was no requirement that they be the same standard set.

Modern day Hinduism is a post-hoc reconstruction. That's not to say Indian religious beliefs and doctrines don't exist -- they do. There are just a wide variety of them and they deserve their own mentions.