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The idea itself isn't so weird, except that this already exists...
It goes like this "Hey, I noticed we're going the same direction and you don't have an umbrella, you're welcome to join me, I'm [state your name]."
You'll probably get some weird looks, but after the initial surprise people could get quite sociable (too bad I'm not).
If things hit it off there's a neat little trick to keep in touch, it's called switching out numbers, but it's a secret technique that not many people know of.
There are a few apps available to help in this regard though, the Contacts app comes to mind, as do WhatsApp and Facebook, or maybe even LinkedIn if you're both on your way to work.
Facebook is particularly easy, just say something like "Hey [name], do you mind if I add you on Facebook?" and they'll usually say "Not at all" because they can't really stop you and it's quite easy to block you afterwards.
I've only once shared an umbrella with strangers which was at a festival where we were all cramped together and I just so happened to have an umbrella (actually it was a friend's, but I was holding it at the time).
While it wasn't particularly awkward, no friends were made that day.
The real challenge is being outside in the rain with an umbrella and finding a stranger without one that's also going in the same direction...