<However, the one thing in common is that they abandoned their old customs, laws and adopted the US ones.

Only, those customs and laws were very similar to those that were "abandoned". And besides, the American experience largely inspired laws and customs in new democracies in Europe like the Constitution in France or Belgium or other countries that removed the kings powers to replace them with parliamentary democracies.
But there is one thing that the US did not abandon (unfortunately, I'm afraid) and that's the influence of religion in society. True, religion was removed from the temporal power, but rather than abandoning it completely, new Americans felt the need to create new ones.
But then, perhaps society wasn't ripe enough to completely remove it from our minds.

Raoul