Let’s look at how you can connect to your servers and group them up using Azure Data Studio.
You’ll see down the left navigation bar. The icon we care about is at the top and will take you to the servers area of the app.

You may as well jump in and connect a server to see what it’s like.

You’ll be given a connection popup. Assuming you can connect using Windows credentials then you’ll just need to put the name of your instance in the server box.

You’ll see the connection appear in your server list. Have a click around, you can see the databases, security and database objects, you’ll be used to these from SSMS.
You’ll be able to connect to all of your servers here, just add them one at a time.
Once you’ve added a few you will probably notice you’ll want to start organising them into folders. Go ahead and add a new server group.

You get to choose a name for the group as well as a description that pops up like a tooltip. You also get to choose a funky colour for it too

Personally, I’ve separated out Live from Dev from QA but do whatever is best in your environment.
If you have instances stacked on the same box then you can create subfolders for these. Just drag and drop folders within folders and instances in those folders.

Look at that, all pretty and organised.