It's unclear, but the tentacles are probably objects
It's unclear if the tentacles are creatures or objects. However, if we look at all the available evidence, we can conclude that they are most likely objects.
1. The tentacles are described as "animated objects".
The rule does say that objects are not animated:
For the purpose of these rules, an object is a discrete, inanimate item
There are two ways to interpret the term "animated object": either it is an object that became a creature (similar to the Animate Objects spell), or it is an object that has the unusual property of being animated.
If the animated tentacle is still an object, then this would be an example of a specific rule (the tentacle is both animated and an object") beating a more general one ("objects are not animated").
I could see it go either way. I'll add that there are already multiple examples of objects that are not inanimate, such as a Rock Gnome's Clockwork Toy, a Dancing Sword magic item, or the hand created by the Bigby's Hand spell.
2. The tentacles do not have complete statistics
According to the monster manual (page 8),
Every monster has six ability scores (Strength, Dexterity, Constitution, Intelligence, Wisdom, and Charisma) and corresponding modifiers.
The tentacles only have AC, HP, Strength, Size, and Damage Immunities. They do not have Dexterity, Constitution, Intelligence, Wisdom, and Charisma scores. This strongly suggests that they are not creatures.
All objects have AC, HP, Size, and Damage Immunities (DMG p.246). Unusually, the tentacles do have a Strength score, but so does the Hand object created by Bigby's Hand.
3. The tentacles do not have a creature type.
According to Jeremy Crawford,
A creature has one of the types listed in the Monster Manual (p. 6–7). A sentient magic item has none of those types. It's an object.
Tomb of Annihilation does not give the tentacles one of the fourteen types: aberration, beast, celestial, construct, dragon, elemental, fey, fiend, giant, humanoid, monstrosity, ooze, plant, or undead.
From Jeremy's tweet, we could conclude that the tentacles are therefore not creatures, but objects.
We could presume that the tentacles are constructs; however this is not at all listed in the book. I think this would be incorrect, just like it would be incorrect to presume that sentient magic items are constructs.
4. The tentacles are not referred as magical in any way
Some argue that the tentacles are animated by magic, in a way similar to the spell Animate Objects. However, there are no reference to the tentacles being in any way magical. They can't be rendered inert through a Dispel Magic or Antimagic Field. This suggests that the animation are mechanical and not magical, and the tentacles move through elaborate mechanical engineering.
5. The tentacles are controlled by the Soulmonger, an object
The tentacles are controlled by the Soulmonger, itself explicitly an object. Presumably, the tentacles are an extension of the Soulmonger. It would make sense for the tentacles to also be objects.
6. The tentacles are not referred as Constructs
If the tentacles were indeed creatures, it would have been so much clearer to describe them as constructs. It would have avoided a lot of ambiguity, and automatically made it clear that they are creatures. The fact that the book uses the nonstandard term "animated object", to me indicates that the tentacles were intended to be objects.