How does this work exactly? Do you explicitly say you're a native speaker and then wait until someone realizes you are not and tell you?

Because I don't see it happening that people will tell you they don't think you're a native speaker organically, in a conversation that's going on about something else.

In my experience, when people think someone else is a foreigner, they ask them where they're from. The assumption that they were talking to an american was violated so they try to determine who they're actually talking to. So if I can manage to not violate that assumption, it must mean that my speech resembles that of an american native speaker.

I'm curious to know what the "mistake" was here. Is it something like referring to in-game items via their name in your native language, instead of English?

So decades later I went a gaming community on the internet and suggested we all play this old game. Everything was fine until I said cooper shield. One guy immediately messaged me "br?" and I was shocked. No one else noticed it. Turns out he was also a foreigner who learned portuguese by playing the game together with brazilians and he recognized that specific brazilian mistake.

This can happen even in our native language.