There are no ethnic Indians like there are ethnic Germans. They can be Punjabi, Baloch, Sindhi, etc.

Short answer: no

Long answer: Kind of, but not in any kind of a way to where a desire to unite is a position with mass popular support. A unified India as a political entity has been historically a rare thing, and mostly exists currently as a unified entity as a direct result of the British Raj. There are several territories and Tribes that seek independence from the Indian Republic. Two of those territories were Pakistan and Bangladesh (FKA Pakistan). There likely are residents of Pakistan who wish for re-unification with India, however per my understanding the prevalent notion is that Pakistan's time within the unified India were not preferred. Ultimately this is a function of that borders can never be drawn with respect for the people who live on the land. There is no line you can draw between any two national cores where everyone who wants to be on one side of the line is on the side they want to be on. To use Poland as an example, in 1931, Lwow had a population of 312,000 people, making it the 4th largest city in Poland. Of those 312,000 people, ~20% self identified as Ukrainian. How do you draw a boundary line between Ukraine and Poland where all Ukrainians are in Ukraine and all Poles are in Poland when a city on their border is so blended?

Countering your example: It's less like Pakistanis are like the Germans in Austria who can't unite with Germany and more like Pakistanis are like the Austrians in Austria who found themselves in Germany after Germany declared them German.

1947 was a long time ago, so I think they consider themselves Pakistani.

There was a religious genocide about it. Now theyre only Indian when they get caught crossing foreign borders in attempts to get better treatment