The sharable link has no connection to the person that will fill in the survey. It is one link for all, it is the "open survey mode" (in contrast to the closed survey mode). You do not have any information stored for the respondents (only if you ask them in the survey). So this link is already "anonymous" if you want to call it that.
The "anonymous mode" that you are referring to is for closed surveys only. Why? Because in closed surveys, which make use of tokens to control who can participate and how often, you usually have stored some kind of information about the participants in the token table (e.g. name, email, etc). If you want to send out a survey to a closed group of respondents, but you still want it to be anonymous, then you use the anonymous mode. This will allow you to send out individual links to respondents, but the token (which identifies each participant) is not stored together with the responses (also the time is not stored), so that you can not connect a response to a specific participant. On a side note: The anonymous mode for closed surveys is not that anonymous at all, as there are still ways to pass information from the token table to the survey and store it in hidden questions, but this is another topic.
If you use a open survey where you just share the link with everyone, you don't know who answered what (as long as you don't ask personal data in the survey that allows to identify participants, but that would circumvent also the measures of the anonymous mode anyway).
So this is already anonymous in terms as far as for the survey tool.
I answer at the LimeSurvey forum in my spare time, I'm not a LimeSurvey GmbH employee.
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