Working as a part-time English teacher as I do, I have met many native Japanese who say they can understand English like a native speaker. Also, I have heard about great teachers who is said to have the ability to do so.
As for me, though it has been about ten years since I set myself to study English arduously, I don't think I am so fluent as them. I don't think I understand English as a native speaker does.
How do I know how they understand English? It is true that you can talk about it with native speakers and you will sometimes get to know how they use and grasp it. But in this way you can only understand specific words or sentences. This just doesn't enable you to be like a native English speaker. I guess what they mean by "like a nateive speaker" is only that they are used to processing English sentences or making themselves understood in English. But it is not evidence that they are like native speakers.