What is a sombrero magnet?

In the United States, people often use the door of the refrigerator as a sort of bulletin board. They will put messages there, reminders to do things--and they are held in place with magnets (the door of the refrigerator is metallic) which are manufactured specifically for this purpose. What the author means is that the friend had brought back such a magnet which was encased in plastic and made to look like a sombrero. The juxtaposition of sombrero and magnet is merely fortuitous--it would be more accurate to say a magnet disguised as a sombrero.

In the image below, you see a refrigerator door to which several magnetized objects have been attached. One of them is someone wearing a sombrero--i didn't immediately find an image of a sombrero magnet, but i'm not going to spend a lot of time on this.

It would be a magnet shaped and painted like a sombrero. There are many shaped magnets, and mostly they're called refrigerator magnets, designed to hold papers and notes to a steel refrigerator.

I don't know why it would be hidden by a menu. Usually, the magnet holds notes to the door, so I would expect it to be on top.

The menu could be a vertically folded leaflet with the magnet on the inner sheet with the outer half folded over it.

Hey, that's right. I was thinking of a full service, tri-fold, plastic laminated menu.

Not over here they're not, that's way too cumbersome; fridge magnet, now that's short and sweet.