By mathematical definition, an octagon is an 8 sided 2-D shape. A stop sign, technically, is not 2-D, so it is not an octagon. A stop sign is 3-D, and has 10 faces. We could consider only the front face of a stop sign, which would be an octagon (now, we are ignoring the thickness of the sign, and it has 8 sides). It is common place to refer to a stop sign as an octagon, because it has 8 edges around it's front face - this is informal language that is fine for day-to-day communication, but is not acceptable in within the precise language of mathematics.
I agree completely with Paul's response but I want to add something. An octagon is an 8 sided 2-D shape but in describing the 3-D "stop sign" shape (I drew one below that is thick enough that you can see the third dimension) mathematicians use the terms vertices, edges and faces.
I have always thought that they do so precisely because the term "side" carries connotations in English that give rise to exactly the ambiguities that you and your colleagues detected.
my 2 cents worth:
Technically, a "side" is not a mathematically precise term, so mathematicians avoid it. However, even we use the term "side" in the same loose way that everyone else does, so really we are looking for a bridging definition that offers more precision for the word "side" without breaking any of its conventional linguistic meanings.
I would propose that a "side" be one dimension less than the object whose side it is. For example, the side of an octagon is 1 dimension (so it is part of a line) since an octagon is a 2 dimensional shape. A side of a cube is a two-dimensional shape (part of a plane).
Curved surfaces still pose a problem and although we could continue to refine our definition, I expect shapes like the möbius strip and the Klein bottle will continue to vex us. Look these up if you are curious about why.
Stephen La Rocque.>