Empircally not true, or at least misleading. Above about $75k/year it is not clear how much MORE happy you'll be, but under that level in a western country you'll be empircally less happy than people at that level. Basically it means if you know your basic needs plus a few extras are in your budget, you'll be happier than struggling to feed your kids and keep a roof over your head.
That ariticle in the Atlantic is an opinion piece based partially on a novel published nearly 90 years ago. 1) A couple things have happened in Denmark in 90 years and 2) My experiences in Denmark have been the opposite, and my Danish friends don't live the lives described in there. My opinion cancels out that writers.
Empirically (again) Danes are very happy because they know they are not subject to poverty due to the whims of fate. Can't say that about very many countries, and the US fails on that measure badly, which is why we as a people are far less happy than the Danes.