I know how to filter the list of tables - using the filter icon. However, this icon is disabled when the Databases node is selected.
Still, I am wondering whether it is possible.
This is now possible if you are using SQL Server Management Studio 2016. You can filter databases by Name, Owner and Creation Date.
I'm using SSMS build number 13.0.15900.1 - you can download a stand-alone version from Microsoft or just use this link: https://msdn.microsoft.com/en-us/library/mt238290.aspx
No, it is not possible to filter this list within SSMS. My recommendation is to write a simple query reaching out to the system catalog view,
select name from sys.databases where name like '%<Your Search String>%';
Or some variation of that query, depending on what information you're looking to pull (I assumed only
name, as the Object Explorer would provide you).