Hi,
Welcome to community.
Yes, it's truely anonymous.
When we use anonymous function in Microsoft Forms to share a form, the admin and creator cannot see user name for the anonymous responses. When they check the responses, it only shows the respondent as Anonymous.
If you go to check the form settings > Who can fill out this form, you will find that there is no Record name option for the Anonymous link type(Anyone can respond), so it won't record user name. Only other 2 sharing types have Record name option.
Screenshots for reference:
Hope above can help you.
Have a nice day and stay safe.
Best Regards,
Tina