It seems you get one and only one psychic blade attack per attack action
You've already quoted the most important part, which is that the psychic blade vanishes immediately after the attack is made. This means that each psychic blade you create is usable for exactly one attack. That, taken together with the fact that you can only manifest 1 blade per attack action, means you only get one psychic blade attack per action. This means that Extra Attack does not directly benefit the psychic blades. However, if you have multiple attacks per action from Extra Attack or some other feature, you are free to draw a real knife and use that for the 2nd attack of your action, and then use your bonus action to make a 3rd attack with your bonus action psychic knife. This actually appears to be a decent way of making a thrown weapon build work with Extra Attack, since you can get your full round of 3 attacks while only needing to draw one weapon per turn (and you don't have to carry a backpack full of daggers).
On the other hand, "one psychic blade attack per action" works directly in your favor when you bring in Haste, Action Surge, or any other effect that grants additional actions. Because you get to manifest a new blade "whenever you take the Attack action", each additional action you take gives you one additional psychic blade to attack with.