There were two ways to find other Yahoo members when using the Yahoo Member Directory, but this feature no longer exists as of February 2015. Due to the member directory not being used and the lack of profile viewing, they decided to get rid of Yahoo profiles.
The nickname and profile picture will not be a feature that Yahoo disposes. Account Settings is where the birthday, other sites linked to the account and the gender of the user will be relocated. Yahoo suggests that if there is any information that the user wants to keep to relocate the information somewhere else.