I tried to dock the snapping panel by double-clicking on top of the panel but it's still not docking with the main QGIS window.
Because you opened it via Project > Snapping options...
Close your floating window, right click on some empty toolbar area and under Toolbars activate Snapping Toolbar as shown below:
But note that the toolbar and the Snapping options partially show the same options, but the Snapping options (rsp. Project snapping settings) are well needed when turning to Advanced Configuration:
In this case you are able to set up snapping properties for each distinct layer:
To dock a floating toolbar
Click on the end of the toolbar (see the circled area in the image below) and drag it to any side of the QGIS window until it snaps into place.
That toolbar will re-appear in the same place the next time you turn it off and back on again.
This allows you to customize the QGIS interface to your personal taste. Personally I like having the Manage Layers and Datasource Manager toolbars on the left side of the screen next to the Layers and Browser panels.