The easy answer is yes, one should hold on to one's cultural identity. The hard question is to what extent one should.

For a country to remain strong, there must be some assimilation as to common language, acceptance of the national legal system, and adoption of the country's history and mores.

Within that framework, one can speak the mother tongue, wear traditional clothing eat traditional food, etc. when amongst themselves. They could also have a grounding in the history of their heritage. They can, and probably should also share those things with society at large.

However, the laws and customs of the current country should trump parochial concerns.

In the US this had been the norm until the last 30 or 40 years. There was a tension between the traditional and assimilation that served the country very well. This, sadly, seems to not be happening quite as rapidly now.