No, but Google keeps careful track of your login history and the location(s) routinely associated with same.
I have no idea why the message text is referring to "this device" but I have received notifications on many occasions when I logged in from a "strange place" on my laptop. If I'm using an ISP that's even somewhat far afield from my home one in my home state I get, "This account was accessed by . . . at date/time," notifications that request that I contact Google and immediately change my password if it was not me. They may be getting more aggressive in trying to prevent unwanted access to user accounts which, while sometimes annoying, isn't really a bid thing.
This has thrown me for a bit of a loop on occasion when I've logged in from someone else's computer (or smartphone using a web browser on the phone) and momentarily forgot that I did so. Luckily I've always recalled after a few seconds and not filed any false-alarm reports with Google or changed my password unnecessarily.
Brian AKA Bri the Tech Guy (website in my user profile) - Windows 10 Home, 64-Bit, Version 1803, Build 17134
In a modern society where everyone thinks their opinion deserves to be heard nothing annoys me more than individuals who mistake their personal preferences for fact.
~ Commenter TheCruyffGurn on the The Guardian website, 8/13/2014