When one uses the Extra Attack class feature, must it be done with only one weapon? If I had, for example, one throwing axe and sword, couldn't I throw the axe and attack with the sword as part of the same "extra attack action"?
Let's examine the PHB before we arrive at an answer:
Other Activity on Your Turn
You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.
If you want to interact with a second object, you need to use your action.
Moving between Attacks
If you take an action that includes more than one weapon attack, you can break up your movement even further by moving between those attacks.
Merging them together, we can say that
No, you need not use the same weapon when making an Attack with the Extra Attack feature
But because of the way Free Interactions work, there are limitations to how you can use each object especially when using said objects would require you to unsheathe or sheathe something. You cannot make an attack (first attack in Attack Action), sheathe the weapon (free Interaction), draw another weapon (second free Interaction), and attack with it(second attack in the Attack Action). That would take two Free Interactions and, as stated above interacting with a second object, will consume an Action.
What you can do, however, is make an attack with a thrown weapon first. This way, you don't need to sheathe the first weapon and can use your free Interaction to draw another weapon then attack with it.
This way, you can throw a Handaxe at a goblin(first attack in an Attack Action), move up to the poor guy -who now presumably has an axe sticking out of its head(movement) and then, drawing your sword(free Interaction), attack him with it(second attack in the Attack Action.
As pointed out in the comments, you could be holding (not using!) both weapons in each hand, attack with one, sheathe it, and attack with the other weapon using the same hand.
It is important, however, that this is distinguished from Two-weapon Fighting
Two-weapon fighting is governed by different rules found in page 195 of the PHB. There are limits to Two-weapon fighting like requiring a Bonus Action and only using Light weapons. If you're using a Handaxe and Shortsword (both light) you can use the Shortsword to attack twice then throw your Handaxe with your Bonus Action!
If you clear up your terminology, your question answers itself. There is no "extra attack action" - there is the "Attack Action" and there are "attacks". The Attack Action allows you to make one or more attacks but there are other circumstances (Bonus Action and Reaction) where you can attack.
Here is what the book says about the Attack Action (PHB p. 102):
The most common action to take in combat is the Attack action, whether you are swinging a sword, firing an arrow from a bow, or brawling with your fists.
With this action, you make one melee or ranged attack. See the “Making an Attack” section for the rules that govern attacks.
Certain features, such as the Extra Attack feature of the fighter, allow you to make more than one attack with this action.
And this is what it starts to say about attacks (PHB pp. 103-5):
Making an Attack
Whether you’re striking with a melee weapon, firing a weapon at range, or making an attack roll as part of a spell, an attack has a simple structure.
... and it goes on and on.
When you roll the dice to hit you are making an attack. There are also things that can replace an attack during the Attack Action, for example, Shoving and Grappling (p. 195).
If you take the attack action then, if you want, each of your attacks can be with a different weapon, or use multiple weapon and Grapple or even multiple weapons and Grapple and Shove if you have enough attacks (but not two Grapples or two Shoves).
For example a 20th level fighter (4 attacks) already holding a shortsword could with the Attack action:
- Shove a creature (Attack No 1)
- Draw your dagger (interact with an object)
- Use your shortsword (Attack No 2)
- Use your dagger (Attack No 3)
- Use your shortsword (Attack No 4)
- Two weapon fight with your dagger (Bonus Action)
There is no reason why an entity need take any extra action or interaction in order to make separate attacks with weapons in different hands pursuant to the multiple attacks granted under an Attack action benefitting from Extra Attack. The Rules As Written require neither that attacks come from the same weapon or hand, nor that an attack made with a weapon in either hand is inherently superior or inferior. It is a common mistake to read into the 5e rules the idea of "handedness" or of a main-hand/off-hand dichotomy. However, nowhere in the attack rules is this dichotomy enshrined.
The nature of attacks following from the Extra Attack feature
To follow the chain of rules and restrictions backwards:
... you can attack twice, instead of once, whenever you make the Attack action on your turn.
... With this action, you make one melee or ranged attack. See the "Making an Attack" section for the rules that govern attacks.
The "Making an Attack" section goes on to detail the different types of attacks and their procedures, but it does not reference the selection of a weapon as having any lasting effect other than on determining the type of attack and the relevant modifiers. This is to say, the attacks are entirely separate, not affected by each other in any way. The closest passage to the effect that there is a main-hand/off-hand dichotomy is in the rules for Two-Weapon Fighting.
Why Two-Weapon Fighting does not imply "handedness"
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.
The Two-Weapon Fighting section describes a condition enabling an additional attack using a bonus action, as long as the triggering attack
- occurs during an attack action, and
- is made with a light melee weapon (as melee or thrown),
and the bonus-action attack
- must be made with another light melee weapon held in a different hand than that holding the weapon used in the triggering attack, and
- cannot benefit from the relevant ability modifier.
As written, it makes no note of the impossibility, otherwise, of attacking with weapons in different hands. Rather, it describes the condition enabling the use of the bonus action. Further, the Rules As Written allow for either hand to be the source of the triggering attack, and for both to be the source over multiple turns, undermining the idea of "handedness."
To illustrate, a character wielding a club in one hand and a sickle in the other may take the Attack action and attack with the club, allowing them to then make a bonus action attack with the sickle. For that bonus action attack, the sickle will not benefit from that character's strength modifier, lending credence to the idea of an "off-hand" attack. However, if, in the following turn, they chose to attack first with the sickle, it would then benefit fully from the character's strength modifier, and were they further to take the allowed bonus action attack with the club, the club would then not benefit, despite that the weapons have not changed hands.
Thusly, the Two-Weapon Fighting rules, properly understood, model something closer to "energy distribution" than to "handedness:" an attack made with a light weapon in a single hand leaves some excess for redistribution to the other hand - just enough to use another light weapon, albeit at somewhat less energy than the first attack - regardless of which hand received the initial impetus.
Compare this to Extra Attack, which has no such caveats. Following the same modeling-analogy, Extra Attack simply reflects a greater allotment of energy, similarly distributable to either hand in full or in part.
(Note: even if including feats, part two of the Dual Wielder feat only modifies the conditions for enabling and using the bonus action, mechanically speaking.)
Since there is no part of the rules that requires either that separate attacks following from a single Attack action need be from the same hand or weapon or that there is any difference between hands with regards to attacks, there is no reason why a character holding two weapons which both lack the Two-Handed property could not use each of them to full effect during an Attack action in separate attacks following from the Extra Attack feature.