eq is function, not an operator. It is called with the form: eq <x> <y> (not <x> eq <y>).
You can fix your template by moving the operands from the the sides of eq to after it:
{{if eq . "login failed"}}
<span class="text-error">Incorrect username or password</span>
{{else if eq . "login success"}}
<span class="text-success">You have successfully logged in!</span>
{{end}}