I logged in for Realms Alpha but the Realms icon is not appearing. Why?
You must have access to Realms Alpha to have the button enabled. If you don't have it, the button will simply disappear. To get it, ask someone with a Realm to invite you to theirs and then you can join their Realm via the button which will only appear if you have access to Realms Alpha.
Source: I tried logging in with my other account without Realms and the button wasn't there.