Because luxuries are spread globally, amenities are "global happieness" by another name. E.h. If you build an amenities building, that still has a global impact because simply means that your luxuries can be spread to another city.
However, there are 2 significant differences:
1. Its much more gradual. In CiV, if you have just one unhappyness you are (apparently there is filter)ed. In CiVI, one unhappieness will mean just one city is unhappy, with the rest sitting on the verge of becoming unhappy. And even being unhappy doesnt have such a drastic effect as in CiV where it completely stopped your pop growth (this is now limited by housing). However I also assume that its going to be much harder to actually gain happieness too.
2. While CiV global happieness significantly favoured tall civs, CiVI amenities significantly favour wide civs. Having lots of cities means that you can use several copies of amenities, and your first 2 pop in each city are free. This means that a civ with 8 cities with 6 pop each (=48 pop) and 2 copies of 2 luxuries needs no additional amenities, while a civ with 4 cities with 12 pop each (=48 pop) and 2 copies of 2 luxuries needs 12 additional amenities from other sources. This is massive.