Google - Weird Desktop/SmartPhone Connection

I was on my smartphone (a Droid) and typed in something in the Google search box. A few hours later I typed the same thing in the search box on my desktop computer. As you probably know, as you type into this box, suggestions appear automatically. Well, the same suggestions happened on my desktop as those on the Droid earlier.

As a concrete example, I brought up Google on my desktop and typed in “Burgundy” and some suggestions (not relevant to the following) appeared. I did this purely to see what would be suggested. Then I went to my smart phone and brought up on my Dolphin browser and typed in “Burgundy mouse hopper” out of the blue. After immediately going back to the desktop and typing in “Burgundy”, you guessed it, “mouse hopper” was now suggested to me.

What the hey is going on here?

Massive, behind-the-scenes syncing based on your Google ID. I assume it’s an aspect of their overall sync service.
I’ve only recently been having trouble with Google on my desktop (past day or so) complaining that it can’t work without certain permissions (cookies, etc.), so I wonder if this is a new thing.