I always get to Wolfram Alpha through the Safari extension, so by the time I get there, I've already entered a query.

Going directly to the web page, I see what you mean. I was already signed in -- via a cookie, no doubt -- but the website still said Wolfram|Alpha until I executed a query.

May be a bug. Probably not a 'feature'.

Wolfram Alpha Pro is a paid subscription service. Once you have subscribed, all you have to do is log in, and you will have the Pro functionality.

The banner will read Wolfram|Alpha Pro.

Depending on your browser settings (etc.) you may stay signed in indefinitely, or have to sign in again periodically.

Note that whether the site is Wolfram|Alpha or Wolfram|Alpha Pro depends on your Wolfram account settings. When you sign in, the page will check with your account to see if you have a subscription.