I use my Harris and Truefitt & Hill soaps directly in the wooden bowl. One of my Harris bowls has been through several cakes now with no ill effect, so it's obviously treated somehow. It does help to use a smaller brush if you're gonna use the soap this way, just to avoid slopping soap over the edge of the bowl. I use a Rooney small style 1 brush from classicshaving and it works great like this.

If you're using a shaving mug you would typically leave your soap in it. Some soaps don't come with their own container and expect you to supply your own mug of some kind. I've got one shaving mug (an Old Spice mug) that has Williams soap in it. If you've got a big brush you may also want to use a mug in preference to the wooden bowls just to avoid any mess.

In both cases you would usually only use the bowl or mug to load up the brush with soap, and you would build the lather on your face. You can also use a separate lathering bowl if you prefer.