At this point I would open it up to see if there’s any kind of receipt/shipping details inside that haven’t been ruined by the weather, and go from there.
If there isn’t any paperwork inside to help you find the rug’s proper home, it’s yours.
Three weeks? That’s excessive and not your fault. If someone left something awful or hideous on your front porch for three weeks, even in error, you’d certainly have done something with it by now. I think your only moral quandary right now is that it might be a nice rug and you might want to keep it…but since you’ve tried to do the right thing repeatedly, give it one more go by looking inside the package for details.
If it ain’t there, you have yourself a rug.