Star Trek governing bodies tend to include the founding species in their name so citizens of “The Klingon Empire” can be called “Klingons” regardless of whether or not they’re actually of the klingon species. The same doesn’t apply to any species like humans who are citizens of The Federation, unless you’re talking about the Mirror-universe Terrans who are citizens of The Terran Empire.
Star Trek governing bodies tend to include the founding species in their name so citizens of “The Klingon Empire” can be called “Klingons” regardless of whether or not they’re actually of the klingon species. The same doesn’t apply to any species like humans who are citizens of The Federation, unless you’re talking about the Mirror-universe Terrans who are citizens of The Terran Empire.
They’re usually named for their home world as well. Klingons are the exception in that their home world used to be Kling, and was then renamed later.
It’s basically the equivalent of Earthlings.