Your Facebook user ID, username, other basic details and the names and ID’s of pages you manage may be stored by Yoast.com for the purpose of debugging and improving the behavior of the application. They will never be used for anything else outside of Facebook and Facebook widgets, nor be sold or given to other people or companies.
Facebook might show your avatar and name on the authentication dialog or other dialogs related to the app. That same line with your avatar and name may be shown on the WordPress SEO plugin pages and in the WordPress SEO admin of your Facebook friends, like this:
A live example: