Does it make sense to say the following?
Yesterday I wore an umbrella and a coat.
Yes. You can wear an umbrella. You can even give a seductive demonic grin when doing so.
I see no reason she could not also be wearing a coat, other than perhaps selling fewer umbrellas for others to wear.
Silly me, someone requested a picture of how you carry an umbrella. I carry mine like this (also with a coat to further illustrate the scenario),
In usual parlance, an umbrella is carried, not worn; but fashion circles speak of wearing an umbrella, when it dominates the get-up. For example, a 2011 manrepeller.com blog entry called "That Time I Wore an Umbrella Indoors" says a recent night "will forever be known (to me, at least) as the night I wore–not held–an umbrella in doors", and fotoblur.com image 91796 is titled "she wore a red umbrella". A 2011 kandeej.com blog entry titled "DIY: 3 way to 'wear' an umbrella" is about wearing part of an umbrella (its canopy) as an accessory.
An umbrella is usually carried. The difference between wearing something and carrying it is that carried items are attached to the body and are not being held in a hand or otherwise. For instance we wear clothing or jewelry because they stay on. If we were to have a ring in the palm of our hand we would be carrying it. So to wear an umbrella would indicate to have it attached to the body. The answers with the umbrella hat shows this. Also having it slung through a belt loop could be called wearing it. In use without a weird gadget it could not be worn
Wear - to carry or have on the body or about the person as a covering, equipment, ornament, or the like.
So yes you can wear an umbrella
I very much agree with the other answers. Though, a child (subclass(?)) of the umbrella is a the nubrella (and I suppose Umbrella Hat falls under that category as well). And that, I believe you very much can wear. So you can wear some umbrellas, but not all. And I suppose it also depends on the context.