This happens when you are currently signed into Blizzard with a different Blizzard account.
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To sign in to Blizzard with the correct account, visit the Blizzard page, sign out and back in again using your correct Blizzard account. After that, go to http://hsreplay.net and sign in by pressing the "LOG IN WITH BLIZZARD" button.

You can also go to HSReplay.net -> Settings -> Connected accounts to verify all the Battle.net accounts you'd like associated with your HSReplay.net account are there.