Somewhere you clicked a “remember me” box at some signin (it may have been inadvertent, not necessarily intentional) and now Google is being helpful, by its lights, in “remembering” who you are. I believe that the only way to disable that feature now – because after all, it is a feature for most users, most of the time – is to delete the cookie that drives it.
And then, as others have already suggested, always and only use the machine – with Google, anyway – as if it were a “public” computer (or incognito) so that it will no longer “remember” you.
If you sign into the computer using a Google account, then that will be part of what’s making this happen.