Around here, we call that da, er, *darn* thing Buckthorn and it is an invasive weed!!!! Get rid of it by any and all means necessary including fire, because once you have it, it WILL spread and choke out *everything*, good plants, weeds, grass, EVERYTHING.
Wild birds and rabbits eat the seeds which then get pooped out of them and that's how the plant spreads.
To get rid of it without burning it, pull it up (and make VERY sure you got ALL the root), put it in a double garbage bag, tie the bag closed, and throw it away. That makes sure the berries can't get spread.
Buckthorn is the single instance where I would use a strong non-organic weedkiller in my garden. It's terrible, 'bout as bad as kudzu up north, and like I said, once you have it, there's no hope -- you won't ever be able to have a buckthorn-free garden again, you will have to move from eradication to management.
Many of the communities around here actually have specific community-building type 'Buckthorn Pulling Days' in which people get together, bring their gardening gloves and hats, and pull and bag the buckthorn.
My condolences, I hope you don't have a ton of those plants!
(And I have no idea if buckthorn is part of the nighshade family, I wouldn't put it past it though!!!)
Whitewater