Most mugs are too narrow. You don't want the brush clanking against the sides -- it annoys you and anyone else within earshot.
^^^ This is why I use wood-handled brushes with my mugs.
I have two actual shaving mugs - one Avon, one generic. Both are Williams/Haslinger sized. Both are flat-bottomed with a slight taper on the sides. Neither have texturing on the inside.
For "normal" sized pucks, I use a regular (large) coffee mug with a flat bottom.