That’s actually the Tweet Button, not @Anywhere. The Tweet Button and @Anywhere, while possible to co-exist with each other, aren’t great matches since the authentication schemes between the two are quite different. The Tweet Button won’t allow users to “log in” to your site, and because there’s no authorization you also cannot obtain personally identifiable information about the end user.
With @Anywhere’s connect flow, you can obtain user information after authorization, and this is detailed in this document section: https://dev.twitter.com/docs/anywhere/welcome#login-signup (specifically, read the part “working with the current user”)