The default rules don’t seem to allow it
The actual effects that took place, those are all quite plausible. The issue is the wizard’s activating the items used by the other characters, and doing it all at once like that.
The wizard in the story speaks a special word, “Lalilulelo” (itself a reference to Metal Gear Solid, for the record), and all of the characters’ magic items activate. That sounds like a command word item, which is basically the default sort of magic item.
First, the rules suggest that you have to be holding the item for it to activate:
A command word can be a real word, but when this is the case, the holder of the item runs the risk of activating the item accidentally by speaking the word in normal conversation.
This isn’t really explicit, but since only the holder has this risk, it would seem that those not holding the item have no chance to activate it. Note, however, that this rule has a whole host of problems, some of which we’re about to run into.
Activating a command word magic item is a standard action
A standard action is something you can (usually) only do once per turn. And it doesn’t say “speaking a command word” takes a standard action, it says the item activation as a whole does. That strongly suggests that you could only activate a single item per action, even if you have several items using the same word. (This is where the “accidental activation” rule runs into serious problems, because if regular speech can trigger an activation, then you should be able to bypass this action requirement since talking is a free action—a huge boost in the power of such items.)
But the rules, especially the rules for creating magic items, can be modified by the DM
Basically, if the DM is allows something, it is allowed. If the player says they want to put safeguards into the items they create that
can be activated even when not holding the items, and
can all be activated simultaneously,
then the DM can allow that if they want to.
And as the aside above about the contradictions between the action requirements on command-word items and the “accidental activation” rule suggests, these rules are already on the vague side. Plus, ultimately, the books only offer guidelines for creating magic items.
Not all items adhere to these formulas directly. The reasons for this are several. First and foremost, these few formulas aren’t enough to truly gauge the exact differences between items. The price of a magic item may be modified based on its actual worth. The formulas only provide a starting point. The pricing of scrolls assumes that, whenever possible, a wizard or cleric created it. Potions and wands follow the formulas exactly. Staffs follow the formulas closely, and other items require at least some judgment calls.
So everything a player creates has to be negotiated with the DM anyway.
In this case, the DM was clearly playing along with the wizard’s player to allow these safeguards to work the way they did in the story.
There is another possible issue with the story
And that is the possibility that combat began in this case. The rules are not as clear as we’d like in this kind of situation, but to my mind, the wizard wanted to take a hostile standard action (activate the items), so this was a combat situation. In order to do what he wanted, he should have had to roll initiative—exactly what he didn’t want to do, because he expected that the barbarian would beat him. The wizard may have had an opportunity to act in the surprise round—allowing him to act before the barbarian or anyone else—but that’s a stretch considering how the rules work. See this answer, and perhaps more notably the chat discussion it prompted, for more details on whether or not this is an issue, and how things (maybe) work.
This is, admittedly, an unclear part of the rules and one commonly ignored—though it tends to run into problems with instant-actioning as the wizard was doing here. Again, DM cooperation makes all the difference.