How to make SSMS Properties Window coincide with object selected in Object Explorer
There's nothing out of the box that will let you do this.

This is kind of confusing... because the Properties window never shows properties about anything in Object Explorer. If you right-click | Properties on anything in Object Explorer, you always get a pop-up dialog box. The Properties window only shows properties for objects that appear in select MDI-style windows, such as query windows.

Having said that, even though there's no way to do this, I'm not even sure it's such a good idea.

In order to return that kind of information to the UI, SSMS has to hit the server with a metadata query. SSMS is a data-driven application; it's just special in that it operates on SQL Server itself. IntelliSense works the same way. If you're on a production server, you probably don't want this kind of thing enabled because every time you touch a different node in Object Explorer (which is very frequently), it will have to hit the server to retrieve the data. And then you get into a debate about how to turn it off, and ......