Part of an ongoing series of 5e (2014) Rules notes. See the end of the post for notes on 5.5e (2024) rules.
Use of a wand in combat is not an Attack, or a Weapon Attack, or anything you could put into the chain of attack actions that a higher level martial character (like a Fighter) can use.
The Magic Missile wand, for example (and other wands use similar language), says:
While holding it, you can use an action to expend 1 or more of its charges to cast the magic missile spell from it.
Using a wand is an Action (more specifically, a Use an Item Action). You get one Action per turn (and one Bonus Action and one Reaction), with very few exceptions.
This doesn’t come into play with the Fighter’s Extra Attack feature at higher levels. While the nomenclature is confusing, that lets a fighter do multiple attacks within a single Attack Action (it isn’t adding to the number of Actions, it’s adding to the, if I can coin the phrase, sub-actions under the Attack sort of Action).
I.e., a higher level Fighter may be able to swing a sword at an opponent three times in a round, but they can still only fire off a wand a single time.
Using a wand also not casting a spell. That is a particular type of action (quite literally, the Casting a Spell Action). If it were, then it would affect the limitations of only one leveled spell cast per turn. If a magic user uses a wand, they can still cast any level spell is cast as a Bonus Action.
The edge case exception here is that Action Surge gives a Fighter an extra Action — which Action could, in fact, be used for Using a [Magic] Item. I.e., getting two shots off the Magic Missile wand.
Wands and 5.5e
Things are mostly the same under the 5.5e (2024) rules, though with slightly different nomenclature.
Using a wand (or any magic item) is done as the newly named Magic Action, as is spellcasting and the like. Unfortunately, the new rules also are quite clear that you cannot use Action Surge to take an extra Magic Action, so we’re back down to a single use of the wand per turn.
On the other hand, allowing use of a wand as a Magic Action still doesn’t forestall using a leveled spell via a Bonus Action or Reaction (5.5e calls them “slotted spells”).