This feature is particularly interesting because a voice remote support page popped up several months ago. Google has been made aware of this and is now testing a new placement for the feature. The feature is available across most popular browsers, including Firefox, Google Chrome, and Opera.
In YouTube's case, the incognito mode will reportedly stop the app from keeping a history of your video searches and what videos you've viewed, and stop the app from making video recommendations and temporarily stop channel subscriptions.
YouTube already has a "Pause Watch History" function in case you don't want YouTube logging your history, but it's a little hard to find (you have to scroll all the way to the bottom and navigate through your history). Rather, it's just something that you'll eventually see in your app in the form of a floating action button.
The feature is slowly rolling out to users of the service's app. The feature will be added without a specific app update, but it is not now available to all users.
There are very few reports of this happening out with the public, so it might still be a closed test within Google. For the meantime, it is advisable to have the latest YouTube TV version installed to increase the chances to receive the new voice feature.