I think that when Islam is exported from its roots, to other countries in which the prevailing culture is not Islamic, some assimilation is typical, but not necessarily inevitable. To some degree, this is going to depend upon the nature of the "host" society, its tolerance for other religions, its political climate, the nature of the prevailing religion or religions, and the nature of the law of the country. Another factor of great significance is the quality of religious observance of any given person of the Islamic faith. In any religion, there are levels of observance, with some people adhering strictly to a faith, while others are not so compliant with its rituals. I am able to comment upon all of this, as it plays out in the United States, the culture with which I am most familiar.
The United States used to consider itself a melting pot, and people were strongly encouraged and usually internally motivated to blend in as much and as quickly as they could. So, historically, I think that those of the Islamic faith, and there were not really all that many in days past, far more Christian Arabs than Muslim Arabs, probably did as other immigrants did and maintained what customs they could in the new world, but gave up some customs in the rush to assimilate. Many of these immigrants were likely to be unaware of the First Amendment and came from countries in which one would never raise or question whether one had one's rights. For example, in times past, I think it is quite unlikely that new immigrants insisted in the workplace upon being able to pray five times daily on a prayer rug. There was no way to maintain a halal diet, which involves a ritual slaughtering of meat and adherence to other rules. Halal butchers were likely to be few and far between. Alcohol, which is prohibited in the Islamic faith, was probably the downfall of many an Islamic teen in the new world, and likely some adults, too. In my late teens and early twenties, I knew many Muslim people of my age who drank alcohol. So, many of the offspring of earlier waves of immigration are likely to be quite assimilated, with the practice of Islam diluted. However, it is amongst that group that jihadists recruit, preying on the notion that assimilation and neglect of Islam is evil and must be corrected by youth.
Prior to 9/11, but after the earliest waves of immigration, the prevailing metaphor for immigrants in the United States began to be more of a salad bowl than a melting pot. I think really this began with the civil rights movement, in which African-Americans embraced their culture and paved the way for other groups to do the same. Thus immigrants began to celebrate their customs and cultures, and this included their religions. Where I live, there are three Hindu temples, all of which were built during this period, as well as at least three mosques. Ethnic food places sprang up all over the place, including many middle-Eastern restaurants. So in this era, with a greater concentration of Muslims, particularly in urban centers, it was easier to maintain religious practices, and there began to be a greater awareness and more laws that reinforced this. Employers are required to accommodate the religious practices of employees, as long as it does not cause undue hardship on the business or on other employees. I would say that during this period, those who wished to be strict practitioners of Islam had the greatest opportunity do so.
Post 9/11, the picture is very different and has two opposing consequences. There are those of the Islamic faith who are trying to assimilate more once again, in order to keep a low profile and avoid the incredible prejudice that has been the result of the attacks on the World Trade Center and the subsequent rise of ISIS and other fanatic Islamic groups. There is no use even trying to explain to some Americans that not every person of the Islamic faith is a terrorist. Some people just cannot hear this. On the opposite end of the spectrum, there are people who become more determined to practice their faith undiluted by the prevailing culture. A common "enemy" can do this to a culture, to be sure, uniting those in it against those who wish to destroy it.
As far as cultural aspects such as music or art are concerned, there are strict rules on both in the Islamic faith. For instance, lyrics about the material world are forbidden and representations of people are forbidden, too. I doubt any Muslim is ever going to be so assimilated that one will be able to find statues in a mosque. As far as singing is concerned, I can imagine western music and lyrics devoted to Allah, with all in a mosque singing in a service, a way that western culture could creep into Islam, but I have yet to see any evidence that this is happening. My guess is that this would take generations and generations to occur.
Finally, throughout any era in history in which one examines this issue, it must be noted that there are Islamic practices that are against the laws of the United States and most other countries. Polygamy is the most significant of these, permitted in Islam and even encouraged in some countries. It is my understanding that this is done quietly amongst some Muslims in the United States, but this is hardly the norm here, and legal authorities can and do intervene. So, there are some aspects of Islamic law that must be forgone if one wants to live in the United States. For those Americans who are worried about Sharia law taking over American law, this is an absurdity. We are far more likely to destroy Islamic culture and religion with our ignorance and prejudice than Islamic culture and religion are likely to destroy our legal system.