On me it had the opposite effect of what was intended:

I opened the website on non anonymous session safari: it asked my name. Then I opened another new non anonymous window on the same browser: it showed my name as expected. I then opened the same browser in incognito mode: it asked my name again. I then opened chrome (non anonymous) and again it asked my name.

Exactly what I expected to see; everything seems to be working as intended. Anonymization online seems to be working perfectly fine.