The best answer, if your connected device is secure from unauthorised use, is to select the "stay signed in" option. That will keep you signed in, until or unless you clear cookies.
It isn't just eBay. I imagine we now all receive endless "sign-in activity" notices from internet businesses. Maybe one day they will refine them so that they only show really suspicious activity. There are so many alerts today that it's self-defeating; I find myself deleting them almost automatically.
