I think information about users should contain information about the country they live in and their mother tongue. Speakers of languages without articles tend to ask a lot of questions about articles. If his or her mother tongue is known such questions can be better understood.
I think this would be a major issue for several reasons.
First, and most simply, there's no way to force people to say where they're from or what their native language is... and there's no way to make them tell the truth.
Yes, it would be helpful but if it's really important for the question, ask in a comment.
There are some other downsides to this...
We have several very capable non-native speakers of English who are as good if not better at helping learners than native English speakers are.
Tagging them as non-native speakers somehow could make newer users less trusting of their answers.
We have several meta questions here about whether it's OK to ask for help from a "native speaker" (no, it's not)... so clearly some people take the word of a native over the word of a non-native speaker, regardless of their fluency.
We are here to form a community of learning and part of that is allowing learners to help other learners and, in doing so, solidify their own knowledge. Forcing (or even encouraging) people to label themselves as non-native speakers could damage that experience.
We want to avoid explanations in the asker's native language
Our site is generally English-only, which is somewhat unique of the language learning sites, which generally allow questions in both the site's language and in English. This helps make the content more useful to everyone. One thing we want to avoid is having people trying to help by explaining the answer in the asker's native language. This is not useful content and will likely be removed (or at least translated). If it is unknown what the asker's native language is, it will reduce the likelihood of non-English comments/answers.
Finally, remember that, while a single person asks a question, answers are for the benefit of everyone. So, while knowing the asker's native language may help address their specific concerns, answers should be broad enough that anyone can gain from them, regardless of their native language.