Carol, that's fine as long as they don't want to change the laws of the land to an alien religion. For people to live their own way, eat their own foods etc is logical and often beneficial to the indigenous population - obvious examples being Food, Indian, Chinese, Vietnamese etc.
My wife was born in Morocco, but is a Pied Noir - French family, so we eat quite a bit of Arabic food - cous-cous etc., and thoroughly enjoy it. Nothing wrong with many Algerian wines either.
This level of integration or addition is very much a good thing - and I suggest an ideal way into understanding at least a little of another culture from a distance. Trying to impose another religion is different though and resented and always will be as History has proven over and over again. Even those who are not religious don't want to be engulfed in overflow laws and values of other cultures.
I clearly remember dinner parties in Australia (the land of immigrants) where we would have people from six or seven different cultural backgrounds which were great together, although I confess we didn't see many black people in those days, with the White Australia Policy, and any Moslems were pretty anonymous as I can't recall any of them either.
Despite these apparent shortcomings Australia had the benefits of no fewer than 160 National groups if I remember correctly and it worked. However, you are correct in saying that they didn't lose their identities, but remained fully aware and grateful of the new life Australia offered, and accepted they were 'New Australians' first and foremost - else why did they go there?
Those who didn't accept this were doomed to be unhappy and eventually packed up and left, none the worse for the experience I suspect, and certainly not missed by those they left behind.
Of course like my wife, quite a few were in those days 'ten quid tourists'. Those who came, saw, and left after the required two years. Nobody saw anything wrong in that - it was 'product sampling', and those who liked stayed to consume. To me this attitude should still apply. Those who come and decide to stay must stay under the terms of the indigenous country, and if they won't or don't want to, then leave.
The creation of the tensions we see currently are because some don't accept this. Yes, I do understand it is largely second-generations who are the biggest element in this across Europe, but why did this never happen in Australia? Although I note that it is starting there as well now with the self-same religious grouping. Can't be accidental can it?
Nationalism is on the rise again, and I am sorry but I remain to be convinced that this is ever a good thing - even if it is probably to ingrained in our tribalism to disappear.