What do we mean by claiming to be British

Have we have lost our ability to claim and wear our ethnicity as a badge to be recognised throughout the world as it used to be when I set off to wander the globe back in the early 60s.
Today the British are drawn from all cultures and many ethnicities and we find it difficult to identify with our own concept of being British, which was formed in our minds when growing up.
One of the problems is that to think such thoughts is deemed racialistic. This fairly recent mode of evaluating and policing a society.  A society which has changed out of all recognition because of the immigration policy which has been implemented over the last 50 years.
There were economic reasons initially to encourage people to come here and of course, rightly, once here it becomes their home and their family, which is raised here, naturally feel that they are British.
Of course it raises the question of where and how the people, who have many generations of established cultural tradition to identify themselves with, feel about this turn around.
It is further muddied by the Politically Correct doctrine which agonises that "to think such thoughts is heresy". This issue of identifying with a culture and especially the norms which are part of that culture, which in the past allowed you to be comfortable in recognising those around you, is now questioned when these norms are demolished and we have a multitude of cultures, each with a different set of standards each vying for supremacy
There is no real substratum to being British any more. Being British is now an exercise in multiculturalism with its multitude of background influences and standards depend on which ever subsection of the society you address.
Their is little commonality and it is made so much more so when segregated by religion.
Religion has defined history. It has distorted the reality which many of us have and to be fair has created a reality for the believer. But it sets up the problematic situation of a society within a society.
The power of "faith" and the insistence that the faith and its "heavenly" instruction takes precedence over earthly rules is a precursor to a clash which eventually numbers will make the difference and decide the outcome !