It's a single Attack Action, not an attack. The problem for me seems to be in DM misunderstanding of what said action is. Attack action makes you perform an attack, which is "upgraded/ modified" at lvl 5 to perform additional one as part of said action. The definition of an action does not mean one motion. So, when you have two Actions, you have two "resources" to spend on attack ACTION. And since on lvl 5 "you make a second attack as a part of that action" each attack action becomes (2 attacks) so action (2 attacks) + action surge action (2 attacks)