This is certainly true of multicultural societies much more than it is true of societies in which the vast majority of people are of one culture. This does not mean that multicultural societies cannot maintain a common culture. It simply means that it is harder for them to do so.
In a multicultural society, there are different cultural groups all of whom want to retain aspects of their culture for at least some amount of time. This leads to difficulties with creating and maintaining a single common culture. For example, in the United States, periodic influxes of people from different cultures have often made “natives” worry that their culture will disappear. This is not a problem for a country like Japan, for instance, in which there is very little immigration.
The US has managed to maintain a common culture, though that culture has been influenced by the cultures of immigrants. However, it has experienced more challenges than a monocultural society would.