The main problem with not being anonymous is that different social circles may clash in a wrong way. I have thoughts that I know would be hardly accepted in many social circles which I may enjoy, why let them have the possibility to form their narrative around me ? I prefer to keep control over what people knows. This allows you to make a specific persona. This is not about being untruthful, but to actually offer what it's of most value in the interaction.
If you maintain the same identity then you have to make a message which will not infuriate any of the groups that are important to you. This limits your life. Anonymity allow self-disclosure, and the possibility of being what it's of most value in all of those groups. Selective disclosure and how you communicate your message are important tools.
I don't have any personal page. I only have a social persona on the Internet under my name for career purposes, littered with some random, but positive, posts, for credibility.
Social networks like Instagram or Facebook have the problem of allowing another persons to form a narrative around my life. I will appear on photos, but I don't have an account myself. I decided not to after I deleted both for the second time. I don't see I lose a lot of social credit because of this, I lose opportunities to start or continue some relationships but I also save myself with having to deal with other persons who I'm not interested in but communicate due to the low cost/risk of the interaction.