Point of clarity: the original claim made by the linked article is that research shows that people with consensual non-monogamous relationships are happier. That is the claim that is addressed here. The question of whether or not they are actually happier is technically vulnerable to a notability challenge, as we have yet to see anyone claim that specifically.
Okay, looking into things a bit, the claim in question is not meaningfully supported by the links it contains. The second link is talking about how willing people are to accept the idea of polyamory, and outside interest, rather than anything having to do with the happiness of people who are polyamorous. The first link is an article in psychology today that explores and unpacks the implications of the second link. The third link is to a list article on alternet that makes a number of vague claims based on "unpublished data" and "yet-to-be-released data", has some "supporting links" that actually link back to the top of the article, and, regardless, never makes any claim as strong as "people in poly relationships are happier". The fourth link is to a study based on a single focus group of polyamorous women that seemed to care a great deal about whether polyamory could be viably combined with feminism, and how this whole thing worked, anyway.
So... at the end of the day, this article is feeding us a load of BS. The various papers linked, or linked at one remove, do suggest that there are some advantages to polyamory (like a reduction of issues with cheating/jealousy) but are nowhere near enough to support the claims that the author is making, and that the author is implying that they support via links.
By all readily available indicators, the article pretends to have support that it does not, in fact, have. If there is research indicating that polyamorous people are happier, it's not presented here. Given the obvious slant of the author in question, the fact that it's not presented here suggests heavily that the author of the article doesn't have it. The article overstates its position pretty significantly, and if the claim is true, it seems likely only so by happenstance.