Yes; that's how the action economy works. Volley is an Action.
Your friend's argument delves into reality simulation, rather than game mechanics.
Because Volley is an Action it can be used as a Reaction. An Extra Attack isn't an Action in this sense; it is something that takes place after your character chooses the Attack action on your own turn. When you are making a Reaction, you are typically acting during another creature's turn.
Beginning at 5th level, you can attack twice, instead of once, when you take the Attack action on your turn. (PHB. 92; Extra Attack, Ranger class feature).
The five elements of the action economy are:
1. Move
Not an issue here.
2. Action
Attack is one choice, Cast a Spell is another, as are Dash, Dodge, etc.
The attack Action is an Action, and so is a Volley. Either of these actions can be used as a Reaction (which takes place during another creature's turn). An extra attack does not say that it triggers on a Reaction. Compare that to this Ranger class feature:
Giant Killer: When a Large or larger creature within 5 feet of you hits or misses with an attack, you can use your reaction to attack that creature immediately after its attack, provided that you can see the creature. {note that this works during the other creature's turn}
Volley has its own constraints (only targets within 10' of a point), in terms of what can be attacked using that Action. Extra Attack has no restriction on how close together the targets must be. (Volley is an Attack with multiple rolls. So is Whirlwind).
3. Reaction.
A ready action uses the character's reaction, which in the case of Volley would preclude an opportunity attack during that same round. (OA also uses reaction). For example, if I used Volley as a ready action/reaction I'd not be able to use the Giant Killer during that same round.
A reaction is an instant response to a trigger of some kind, which can occur on your turn or on someone else’s. The opportunity attack, described later in this chapter, is the most common type of reaction.
Ready
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn. First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it.
4. Bonus Action.
Not an issue here.
5. Interact with an object.
Not an issue here.
These game mechanics, or gamisms, are put in place to avoid things like my 11th level Fighter, Champion, getting his Action (three attacks) and also getting to use his reaction for an Opportunity Attack that gets three attacks rather than one.
