This is where it gets tricky. I know some Americans ( which includes Indian, Mexican, El Salvadorians.....) who made their life in US ,they worked so hard for that passport. Us has given them everything in their life so they like to address themselves as an American than an Indian or Mexican or what ever.
Now there is another case which I have seen especially among Indian they like to brag that they are an American /Canadian /...I have seen Indians trying to be American by imitating accents,watching baseball which they don't even know what it is.
The 3rd scenario is related to the 1st one I know only about India so not commenting on others. I assume it is a part of cultural integration. I was almost a vegetarian before coming to US now I eat ham,beef,fish what not,I used to be religious now I am not .So people change and no one chose to be born in their home land but they chose these nations.There is no shame in being an indian or any other nationality because it was not anyone's choice.
Personally, here I might be wrong from others perspective if a country is letting you in ,giving a place to live allow you to enjoy all the benifits of that nation (like powerful passport, free education, rights to start a business, free health care.....care.....-I know there are flaws in all of this, but still it is better that where you are born) and you refuse to even acknowledge as a part them I don't think that is very cool .But showing off in FB group that is BS
It is a matter of perspective