I can see how being a big part of society would exert pressure to realign with society's default approach towards anonymity (1) but I don't see how that would imply any final status of online anonymity. Other forces might exert pressure in the other direction. The technology itself might resist some pressure. Access to certain technologies changes the societies themselves.

(1) And of course society in general used to be and still is pretty anonymous. I can easily buy a newspaper in almost perfect anonymity through regular channels, apparently I need to take special precautions to get the same status online.