Additional social icons/links in user/author box?

By default, core WP has no fields for any social links. Any that you see are there because of your theme or a plugin. There is however, a field for a user’s website. It’s odd you say there is none. The website field should be visible in any user’s profile, part of their contact info.

As for the other social fields, determine what is responsible for the fields you do have. Switch themes or deactivate plugins until the fields disappear. Whatever you did last will be the one responsible. Through that module’s dedicated support channel, you can then try requesting additional fields be added; or abandon the responsible module and switch to one that does offer all the fields you want. Be aware that any existing data saved by a plugin is unlikely to be compatible with another, so existing users will likely need to re-enter their social data.

It is possible to add any social fields you want with custom coding, but it would require reasonable PHP coding skills to implement something this way.