As far as I'm aware, there is no public API exposure of a browser's default homepage/search provider. So how does Google know to display this? It only comes around when Google's not the default homepage / default search provider on my browser.
I can only assume they're inferring from numerous variables, such as the referrer. I wasn't able to successfully dig down into Google's compiled JavaScript. I'm not even sure if it's detected client-side or server-side.
I'm on Firefox 44, but I've seen these banners on Chrome, too.