The rules state:

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.

Would the bolded parts apply to the extra attack as well as the triggering attack to allow a second weapon to be drawn?

One free object interaction per turn

You can also interact with one object or feature of the environment for free, during either your move or your action. [...] If you want to interact with a second object, you need to use your action. [emphasis mine]

You can interact with one object either during action or movement. If you want to do interact with two objects you have to use your action to interact with the other. If you haven't interacted with an object then you can do so anytime during your action or movement including between attacks. You will note that your extra attack is part of the attack action, but it isn't a second action.

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

That is both attacks together are a single Attack action. You can't give up an attack to draw a second weapon, you'd have to give up both. You don't magically get two free object interactions. If you want to draw two weapons as a free action, you'd have to take a feat like Dual Wielder which has this ability:

You can draw or stow two one-handed weapons when you would normally be able to draw or stow only one.

A key part of the rule there is in the first sentence.

You can also interact with one object or feature of the environment for free

By this, assuming the character has not had their one free interaction, then yes. If they have however, the feat Dual Wielder has this:

You can draw or stow two one-handed weapons when you would normally be
able to draw or stow only one.

You can only draw one weapon per turn for free

The rules state:

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.

When you take the Attack action, you can draw a single weapon as part of that same action. Note how the rules specifically says "one object" and "your weapon". Also the sentence directly following this passage makes this more explicit:

If you want to interact with a second object, you need to use your
action. Some magic items and other special objects always require an
action to use, as stated in their descriptions.

Drawing one weapon is one interaction with one object. Trying to draw the second weapon would be a second interaction with a second object and thus it would require another action to perform.

Note that Extra Attack only adds attacks to a single Attack action, but the rule do not care how many attacks you are making, because you can only take one free interaction per action and only one per turn.