Familiar is a very loosely based word. When companies like Google say familiar they're normally talking about artificial intelligence deciding it is confident enough that it has seen enough patterns to decide that the machine you're using right then is one that they have seen you use before.
Do keep in mind that using Google Chrome for Google is the same as using Microsoft windows for Microsoft. It's literally their own little Disneyland where they can do basically whatever they want. Because of that, they get access to things that arent typical browser methods that they're able to get your information from. Sometimes it's a browser signature or sometimes it's your IP addresses coordinates.
I don't think they would release this information specifically because it pushes the creepy factor but in terms of computer science it's a tremendous revolution. I could verify you with patterns such as your movement patterns of the mouse, how your typing patterns look (forget keylogging, the speed of keys typed is enough to be unique), how you scroll through the page, where you like to hover the mouse when you're reading, there are a lot of ways to watch someone that make it so if you combine them all together it really does make you an individual who can be identified in a crowd.
If you go to
chrome://flags you'll get a little better of a perspective of the information that is collected within the browser besides just the cookies. In the histograms section, you'll find analytical data towards the satisfaction of the searches that you made and all sorts of other data points that would make a data scientist smile.
Long story short, artificial intelligence and data science have grown far beyond cookies.