Which WordPress widget will allow me to display categories like Blogger?

I'm not aware of any such plugin or widget. You might notice that, if you are a prolific writer, the list can grow really, really long and add to the page load time and experience, however slightly. Now, there are two ways to create something like that. One is to get it coded via a developer. The second way is to go to dashboard>>appearance>>menus. Now, add custom links for the dates (you won't get the count alongside the months. Then, add the posts manually under each date. Next, in the widget area, add a custom menu widget and set it to this menu. Here, as well, you'd either need a javascript coder to add the counts and make it collapsible or DIY via some basic jQuery. However, like I said, I'm not aware of any such plugin/widget. Someone else here might. I didn't google it because I'm assuming that you must have already done that before posting this question. If not, please do that ASAP.

I don't recommend you actually do that in your sidebar. If you write with any frequency, you'll take up the entire sidebar with links once the reader clicks a month. There's no good reason to NOT require them to click the month to arrive at a separate page.

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