This gives me a lot of feelings, but i think if you’re trying to convince a Big Bad that you are an agent against Asgard, or at least against Thor, You would say you were from somewhere else too. Kurse’s remembers him from the prisions, Loki is trying to play on that to earn their trust. Malekith wouldn’t know Loki was Jotun by birth, would he even know Jotuneheim? What did it look like 5,000 years ago? He certainly doesn’t know what it looks like now. Loki Looks Asgardian, so he has to assert his Otherness to get them to trust him. Its kind of the easiest way to do it.