Google added Save button on the local business map results for desktop devices as well now. It was first added to the results on mobile devices last year. The functionality and purpose of the button are the same. You hit the ‘Save’ button on a map result to get a few pre-populated lists where you can add the particular map result for future reference.
Here’s how it appears if you have not spotted it yet:
Once you add the map listing to the available or a new list, you can check it back from the ‘Your Places’ section in your map account. It appears something like the image below: