An "anonymous" questionnaire is not "truly" anonymous. There are data tracks that can allow someone with access to the database to determine who responded. However, an anonymous questionnaire will not reveal who submitted the response through any of the standard UI elements. So, a teacher in a course cannot see who the respondents are if they are reviewing the responses. And, if a questionnaire has been set to "anonymous", and responses have been submitted, it cannot be set to be "not anonymous".
Information identifying the respondent with the response is still stored in the database. This allows an anonymous questionnaire to still allow "one response per user" for example. Data still exists, identifying the user in the "questionnaire_attempts" and "questionnaire_response" tables as well as the Moodle logs. But, you would need to access the database via mechanisms other than the questionnaire UI to extract that information.
I have created issue CONTRIB-6135 to discuss the possible need for more anonymity.