Does the tentacle rod prevent multiattack?

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?