I can understand some of the points you make, yet I still see people saying Azad Kashmir is not a part of Pakistan. I don't know why they call themselves Kashmiri when they don't speak languages like Koshur, it is all quite confusing to me. Are you from AJK?
For the sake of clarity, it is important to understand the context of each situation, so I'll try my best to explain.
The Soviet Union consisted of 15 republics, but in the West, they were usually just referred to as Russians. It’s a colloquial simplification. Such situations exist in different forms around the world. But every situation is different and develops through local processes.
In the Indian state of Bihar, pretty much everyone identifies as Bihari because of a historically shared experience and centuries of somewhat unified rule as a Bihari grouping. But they do not speak the same language. The main language was Urdu, now largely spoken by Muslims, and Hindi, which replaced Urdu because the Hindu community did not want to use it. Other languages such as Bhojpuri and Maithili, among others, also exist.
I have heard the language point used mostly by Indians to differentiate the people of Kashmir and their associations with each other. For a country that uses terms like “diversity is unity” to justify its existence, it’s a bit rich to question the identity of the Kashmiri people.
With regard to Jammu and Kashmir, it has many different regions, but the most dominant identity is Kashmiri. Over time, usage of the term “Kashmiri” has become part of people's identity, even if they do not speak the language of the Vale.
It needs to be understood that identity is never static; it is always evolving. Different experiences at different times affect the identity of an individual or a group as a whole. This is only natural, and the same is true in Azad Kashmir. While most do not speak the language of the Valley itself, they strongly adhere to a Kashmiri identity. This was a natural process due to mutual suffering and literally centuries of living in close proximity under broadly similar governing structures. Religion also played a role in developing a broad Kashmiri identity.
As an example, in Pakistan, Shalwar Kameez was not a common dress across the country. At independence in 1947, people wore different local dress across Pakistan. But today, Shalwar Kameez is just about the only local dress worn throughout the country. People have adopted it as part of their unified Pakistani identity. It was a natural process. The same can be said for Urdu.
The core issue is identity, not only language, although language was a central component of identity in pre-modern times. People feel they are Kashmiri due to various factors. Why shouldanyone question that? If they do, then we can lay open the identity of everyone, and no one will escape without harm to their pride.
Most Pakistanis in the UK are from Azad Kashmir, but they strongly identify with the Pakistani identity while being proud of their Kashmiri heritage, both can co-exist, it’s not unusual. These are the facts on the ground. Why should anyone question it? If anyone does, then let us open their claims to their own identity. I can assure you it won’t be a pretty sight.
They are Kashmiris due to the development of natural historical processes. That’s how every single identity in the world has evolved and developed. It’s nothing unusual, it’s a natural process.