I don't think you can directly organize the service in the browser as they are referenced by service type (WMS, WFS, ...) then by connection to a specific server.
One workaround is to add each of the service you want to use to a QGIS document and save each of them as a Layer Definition File (.qlr Export> Save as Layer Definition File...). (note that .qlr is an XML file that contains a pointer to the layer data source and the style information, this will not create a local copy of the data)
You may now organise all your .qlr in a folder structure organised by area and/or topics and access them in the browser (you could add the folder as favorite for quick access).
While this could be a lot of work initially it has the added advantage to let you name each web service as you like rather than using the server name on top as letting you choosing how you organise them.
Also this way of organizing data is not limited to web service and you may choose to get all your data, regardless of data type, format, and storage location, saved as .qlr and organised in an arborescence that suit your specific need (If you go this way I suggest to have a naming convention and/or folder organisation that let you easily determine the kind of data (raster or vector, point, line or polygon...)