For the most part, it depends on what you'd call "official".
afaik, SRB2Kart is developed by a team outside of Sonic Team Jr known as Kart Krew, as you may already know. Although this automatically makes it unofficial, the SRB2 Message Board does still have direct integration for it. (i.e. it gets its own section in Addons & More), and is otherwise completely endorsed by Sonic Team Jr.
On a more technical note, the EXE used to run SRB2Kart is actually based on SRB2 2.1, and it's not even all that different. However, the dependency files for the two games are completely different.
So, it's sort of neither, and sort of both, at the same time.
A mod that happens to be worked on by many people who also work on main SRB2.
The distinction between "official content" and "fan content" is kind of weird when the whole thing is a fan project in the first place, but ultimately the only things that are official offical is the stuff that you can download directly from https://www.srb2.org/
Kart is a mod of SRB2 and isn't made by the same team that made SRB2. Therefore, it's an unofficial mod. Whether or not it's supported on the Message Board doesn't matter, it's made by a different group.
For those wondering what an official mod would look like, that would be something like Bonus Characters, a collection of characters made by Kart Krew that couldn't be in Kart because of how the game was designed