The tentacle rod's description includes the following (emphasis added):
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.
Answering this recent question, I concluded that the three-attack feature of the tentacle rod is incompatible with extra attacks from its user. In this older question, it was concluded that it is also incompatible with opportunity attacks.
Usually, magic weapons have the respective item category (e.g. the scimitar of speed, which says "weapon (scimitar)") that defines damage based on those weapon types detailed in the PHB (p. 149). The description clearly states that the rod is a weapon (vide infra), its category, however, is rod. Rods, contrary to staffs, cannot usually be used as quarterstaffs (see this question about rod of the pact keeper).
It seems, therefore, that the tentacle rod is a weapon by RAW but has no weapon type or weapon damage defined.
The rules on improvised weapons (PHB pp. 147-148) state:
An improvised weapon includes any weapon you can wield in one or two hands, such as broken glass, a table leg, a wagon wheel, or a dead goblin. Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club.
It further states:
An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object).
Now the last part should not apply because the rod clearly is a weapon. I can simply decide what weapon the rod is like (probably a quarterstaff, club, or a whip).
I could not find conclusive information on what the damage would be when making a normal weapon attack with a tentacle rod.
Is there anything more clear on the subject of the amount of damage, either additional rules on improvised weapons, or specific information on the tentacle rod, that would give guidance on the matter?