When you're able to log into your ESO account on our website, but you get a login error when you try to log into the game, it's because you do not own the game on the account you're logging in with.
You can confirm this by looking at the Account Summary on your account page:
Account Summary should be highlighted on the left.
Your account status is on the right, if it states "You have not yet made any game purchases..." as in the image below, no game code has been added to this ESO account. You can click the Buy Game button to purchase the game.
You can contact customer support if you need assistance with finding your game account.