For races, there is no conflict.
As far as the updated race entries go, the WotC products page for the book explains:
Each of these peoples represents a race option when you create your D&D character, expanding on the choices in the Player’s Handbook.
So there's no conflict; I would expect that in the same way that you can play a PHB "Dragonborn" or a "Metallic Dragonborn" from Fizban's, the Volo's Kobold and the Multiverse Kobold will just be two different options for playing a kobold character.
The PC race options are, as I understand it, intended to be more along the lines of the custom lineages from Tasha's, where they provide stat bonuses that can go wherever you like them and make some usability fixes. In some cases, the updated options do represent a sort of quasi-errata, like how the Dragonborn options in Fizban's allow you to use your breath weapon as part of an attack action instead of as an action, which makes it actually useful at higher levels (where the opportunity cost of not using weapon attacks made it undesirable for melee combat experts).
Errata is fine for correcting actual mistakes or clarifying text that didn't quite say what it meant, but rebalancing is really another thing entirely, especially for character-facing rules. It's often better to print a superior version of a thing as an "option" rather than try to say that everyone is required to go back and change their characters because a new version of the Ranger came out or whatever (potentially invalidating existing character choices). This is especially true where it come into contact with AL campaigns and so on; it's just easier to leave the 8-year-old information alone and print a new "option" instead.
Monsters might get errata, but who can say?
Updating monster stat blocks isn't as big a problem as updating player-facing information, but it's still potentially making a big mess.
According to the product page, the book includes
Updates to the monsters include making spellcasters easier for Dungeon Masters to run; giving many monsters more damage and resilience; and improving the organization of the stat blocks themselves.
On the one hand, I wouldn't be surprised if they released an update document to tell DMs how to update Volo's and Mordenkainen's monsters to fit with what's in Multiverse. But on the other hand, such a document would be nigh-unusable in actual table play, wouldn't provide the stat block organization benefits, and potentially sucks the wind out of the sails (or should that be sales?) in terms of moving Monsters of the Multiverse as a product.
A document that makes DMs say "I don't really need that new book, since I have all the information right here" and at the same time gives them a bad experience when trying to build and run a game... well, that's maybe not a good thing to release.
