Huh, I'm not sure why you're seeing notifications from an application you haven't opened. As far as I know, that shouldn't happen. Still, upward and onward.
Your Mac is doing what it's supposed to. You have a couple of options. You could locate the Tweetbot application, Control (right) click on it, choose Open, and you'll be asked if you're sure you want to open it. Click Open and it does. Gatekeeper asks you just a single time if you'd like to open an otherwise forbidden application. Once you do, you'll never see that warning for that particular application again.
Or--and this is something niftyish that I just discovered--hold down the Option key and click on one of Tweetbot's notifications. Tweetbot opens without any annoying "Are you sure?" dialog box. (And will continue to open without bugging you forever more.) I haven't been able to confirm that this works with any unopened application that shoves notifications into Notification Center (nor, again, am I sure how those notifications got there in the first place if you haven't opened the application), but if it does, it's a handy shortcut in such odd cases.
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