Reasonably, the original hit dice would be replaced by class hit dice
I believe an appropriate reading of this section would imply that a 1st level Kobold Sidekick would have 1 hit die of 1d6. I base this on the generalization that Challenge Rating is not the same as Level, and to accept a monster's existing hit points as its starting hit points leads to absurd scenarios where a sidekick is the most powerful creature in a party.
Firstly, we can't inherently assume that every monster of CR 1/2, or any other CR, is a a uniform "base level" when we start applying sidekick features. As an example of this, a Monodrone Modron (MM p.224) is also a CR 1/8 creature, like a Kobold. However, the Monodrone only has 1 hit die of 1d8. Similarly, most CR 1/2 creatures have around 3 hit dice of various sizes, but Orcs (MM p.246) have only 2d8, a number of creatures have 4 or 5, and the Satyr (MM p.267) has the most with a full 7d8.
Secondly, the Dungeon Master's Guide advises that a single monster of a CR equal to the party's level poses a reasonable but not deadly challenge, and many other references throughout the DMG and the Monster Manual make similar references to that assumption. So we can't use CR as a 1:1 equivalence to "level" by assuming something like "CR 1/2 or lower creatures are level 1 as built," which would lead to the unusual scenario that a Kobold has a single "hit die" consisting of two dice. Instead, we should simply use a singular instance of whatever die is in the monster's hit point formula (a Kobold showing "2d6-2" hit points therefore has a 1d6 hit die).
Thirdly, Sorcerers and Wizards who would choose to put only an 8 in their Constitution would likewise enter 1st level with only one hit die of 1d6. The difference is that these character classes explicitly say you gain maximum hitpoints for the first hit die, meaning they would start with 5 hit points (1d6-1, maximized). That maximum hit point line is missing from the construction of a sidekick character, so its hit points should be the average (here, 3, or 1d6-1 taking the average roll of 4). I bring this up to point out that such a low number of hit points is far from unusual for 1st level characters that aren't explicitly built to survive a single hit.
Side Note: Tasha's does technically say you should roll for hit points on additional levels for side kicks, but taking the average is an accepted choice almost everywhere else when adding hit points, so I don't believe it's unreasonable to do the same for sidekicks. Similarly, if a sidekick is to be use as a player's primary character, taking the maximum for the first hit die is also a reasonable modification.
Lastly - as Tasha's Cauldron of Everything makes clear much earlier in the book in the Customizing Your Origin section - what is published carries with it "certain cultural assumptions." As a DM creating a Kobold Sidekick, you are now changing those "cultural assumptions" as well by allowing a monster such as a kobold, literally described as "easy prey for predators" and who "serve [evil dragons] as minions and toadies" (MM p.195), to receive a pseudo-adventurer's training and ability. The resulting change in hit points can be explained however you want it to be, in character or out, if you even need to, since it is hardly the most dramatic change the creature is undergoing:
- As character development: The more rigorous training it received in non-combat abilities means it is no longer as focused on life-or-death survival, and it's lost some of that drive to fight to the very last breath as a result.
- As character balance: Monsters normally don't get death saving throws, so a slight increase in their HP balances that out. Now that the monster is eligible for death saving throws, it should lose those additional hit points.
- As a counter to absurd situations: "You are not getting a Gnoll Sidekick who has 22 (5d8) hit points at first level."