Not really. On top of all local traditions of each nation, there is an overall Christian/Catholic culture that is common.
Multi-culturalism is also about knowing local customs of many cultures that may include rites that are not morally acceptable in your own culture. Example given - a Muslim having more than one wife, Hindu society divided by casts, Roma people living as vagrants as a custom not need etc.
In the Church even if say an African couple may have an underage spouse because it is a local custom to marry younger, you still cannot break certain cultural rules that are set above the custom and marry more than one wife in a Christian rite, or believe that the casts are divinely imposed etc.
The multiculturalism in EU is really about not commenting on others cultures as “the European way” of allowing anyone to live as they please.
However when they banned burkhas in Austria and other places, is EU really multi-cultural or is that just propaganda to present an invented 'European way", as a super-culture to be imposed on all other cultures included in it (local European cultures and migrant cultures)? Because once they ban an aspect in a culture multiculturalism is gone. For example they banned crosses at work in UK in order to “not offend the Muslim co-workers”. What? The Muslims do not care about crosses, the only cares about religious signs being too religious is the Brussels Government that leans towards socialism and is using any conflict to its advantage.