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Made by the drow, this rod is a magic weapon that ends in three rubbery tentacles. While holding the rod, you can use an action to direct each tentacle to attack a creature you can see within 15 feet of you. Each tentacle makes a melee attack roll with a +9 bonus. On a hit, the tentacle deals 1d6 bludgeoning damage. If you hit a target with all three tentacles, it must make a DC 15 Constitution saving throw. On a failure, the creature's speed is halved, it has disadvantage on Dexterity saving throws, and it can't use reactions for 1 minute.
Moreover, on each of its turns, it can take either an action or a bonus action, but not both. At the end of each of its turns, it can repeat the saving throw, ending the effect on itself on a success.

My question is whether this effect would prevent a creature from performing a multi-attack.

I'm playing in D&D 5e and the tentacle rod has a similar effect to the slow spell, but the slow spell explicitly states that you cannot make more than one melee or ranged attack. It does, however, say that you can perform either an action or a bonus action but not both on your turn.

Some applied examples:

1. An Umber Hulk has multi-attack where it attacks with claws twice, and mandibles once. These could be considered all part of a single attack action, or an attack plus a bonus action.


2. A Bandit Captain has multi-attack where the MM basically takes dual wielding and combines it into a single action. The second attack would normally take a bonus action from a PC, but no such statement is made for the Bandit Captain.

In either or both of these scenarios should the tentacle rod prevent any additional attacks beyond the first, or is the creature's action economy unaffected where a PC's action economy would be?

Multi-attack is an action, so tentacle rod would not prevent multi-attack.

Players get multiattack too (of sorts), not just monsters. The Fighter gets Extra Attack at 5th level. The class feature description states:

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn. (from the PHB, pg 72)

Multiattack works the same way for monsters as Extra Attack does for players (in terms of action economy). It only takes one action and is not split up between the action and bonus action. That's why the game is very easy to run. You get very general rules that apply to many instances of the game. Only if there is an explicitly stated exception to a rule do you break it.

Regarding example 2 from your question:

A Bandit Captain has multi-attack where the MM basically takes dual wielding and combines it into a single action. The second attack would normally take a bonus action from a PC, but no such statement is made for the Bandit Captain. (from your question)

Monsters and player characters are designed differently. A monster is generally stronger than a PC in certain ways, but PCs have the advantage of not being run by one person (among other benefits). That being said, monsters can have certain attack options that PCs cannot.

On a failure, ... it can't use reactions for 1 minute. Moreover, on each of its turns, it can take either an action or a bonus action, but not both.

The Tentacle Rod does have some cool effects. If the monster has a use for its bonus action like casting healing word or using spiritual weapon, then it could not do those actions on the same turn it attacks, because they use bonus actions. In the same way, an evil wizard could not use counterspell because it is a reaction. However, Multiattack counts as one action and is not stopped or split up.