Extra Attack only modifies a regular Attack action
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
I think the fact that "Attack" is capitalized seems to implicate the Attack action under actions in combat vs. any attack.
Horde Breaker could be used in conjunction with Whirlwind Attack
Once on each of your turns when you make a weapon attack, you can make another attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.
Since Horde Breaker triggers off a "weapon attack" not an "Attack action" you should be able to use Horde Breaker to target one of the enemies next to you. Its simply triggering off one of the attacks made as part of the whirlwind.
Optimization breakdown
Since all Hunters get either Volley or Whirlwind, the Extra Attack class feature is there as a stopgap in the meantime and also a fallback for when there is only one enemy in range to be targeted. Horde Breaker should still be viable even though its utility will diminish in comparison upon reaching level 11 as a ranger, but even then its still one more attack per turn.
3 attacks on your turn (if you don't use Action Surge).
(It could go up to five with Action Surge, though.)
The Two-Weapon Fighting rule states (PHB, page 195):
When you take the Attack action and attack with a light melee weapon
that you’re holding in one hand, you can use a bonus action to attack
with a different light melee weapon that you’re holding in the other
hand. You don’t add your ability modifier to the damage of the bonus
attack, unless that modifier is negative. If either weapon has the
thrown property, you can throw the weapon, instead of making a melee
attack with it.
The bonus action used as part of Two-Weapon Fighting is an attack, but it is not the Attack action.
When you are a 5th-level fighter and you take the Attack action, you can make two attacks. You can also use your bonus action to make an additional attack. Notably, you can only take one bonus action per turn, since the rules for bonus actions state:
You can take only one bonus action on your turn, so you must choose which bonus action to use when you have more than one available.
You only add your ability modifier to the damage of that bonus-action attack if you have the Two-Weapon Fighting fighting style (PHB, page 72):
When you engage in two-weapon fighting, you can add your ability
modifier to the damage of the second attack.
Therefore, you are taking the Attack action (2 attacks) and then making an additional attack with your bonus action (1 attack), for a total of 3 attacks if you're using your action plus your bonus action. There's a clear difference between them.
Action Surge lets a 5th-level fighter attack up to 5 times in one turn, once per short rest.
The description of the Action Surge feature says (PHB, page 72):
Starting at 2nd level, you can push yourself beyond your normal limits
for a moment. On your turn, you can take one additional action.
Once you use this feature, you must finish a short or long rest before
you can use it again. Starting at 17th level, you can use it twice
before a rest, but only once on the same turn.
Which means that if you use Action Surge, you would have 2 actions and 1 bonus action on your turn.
So with this, you can take your Attack action (2 attacks), then use Action Surge and take another Attack action (2 more attacks), and finally your Two-Weapon Fighting bonus action (one more attack), for a total of 5 attacks during your turn.
Best Answer
No.
You quoted the relevant text. Extra attack specifically only triggers when the Attack action is taken, not on any attack. By contrast, the Multiattack racial feature is an action that allows you to take three attacks.
Thus, because the Multiattack action is not equivalent to the Attack action, Extra Attack does not trigger. Additionally, you do not get two Multiattacks, as is described here.