Your profile is always public, but you decide what it includes.
Among all the information you can include:
Profile picture
Username
Bio
Location
Website
Your portfolios
Your watchlists
Only the username is mandatory, and you can choose any username you want, as long as it is not already taken by another user.
In short, it is entirely possible to never be personally identified, even with a public profile.
Here is an example of a public profile of a user that is completely anonymous:
Change my profile information (photo, username, bio…)
Change my profile information (photo, username, bio…)