Can we add a text/property showing next to the avatar? That would be native/non-native speaker so that we can give more tolerance to non-native speaker like me. I always think that while asking about meaning on a particular phrase/word, some users might think "What kind of stupid question is this?"
You can always report in your user profile that you are not a native English speaker.
I don't think there is another SE site where being a native English speaker is relevant. Maybe it can make sense to report on an SE site about a language that you are not a native speaker of that language, but reporting that doesn't make sense to report that for sites like Stack Overflow, and Super User.
There are some problems I see with this such field. As the profile can be copied to another SE site, what would "I am not a native speaker" mean in other SE sites? If I am on the Italian Language and Usage site, what would it mean? Would it be relevant for that site that you are not a native English speaker?
Considering users that are native speakers of two different languages, I see difficult to catch all the possibilities in a field.
Then, there must be a reason, if a similar field is not already set in the user profiles.