I just installed Google Chrome on a fresh Windows 10 VM to repeat the experiment outside of Incognito Mode. First, I opened a video. Then, I checked the front page and confirmed that related videos had been added. Then, I closed Chrome, reset the VM's state to before I had visited YouTube, and checked the front page again. (I don't think this scenario would be too indicative of a prying user; I can easily imagine corporate systems that regularly wipe out browser data.) Again, the related videos were replaced with generic videos.
I then tried the same thing, except by clearing browsing data through Chrome's UI instead of resetting the VM. The results were the same.
Obviously, these observations could all have been manipulated by sufficiently conspiratorial fingerprinting. But the simplest explanation is that the site uses and respects browser cookies for its recommendations, in the absence of good evidence to the contrary. Thus my request.