My brother collects Old Spice and has a LOT of mugs.
In general, you should shoot for about 2-1/2 to 2-3/4 high and 3-1/4 to 3-1/2 wide. A small flair from the bottom to the top of about 1/8”. Standard soap is about 2-1/4” diameter.
Since yours is wood you may need vary the diameter some to make up for the thicker wood.
You can check out different styles here….
They who can give up essential liberty to obtain a little temporary safety, deserve neither liberty nor safety. Benjamin Franklin