As you can see on this website the Tweeter share exists on top and on bottom, also all posts have tweeter share buttons individually.
It looks like you have filtered cookies for wpforo.com. But external cookies are stored under their corresponding domain (but still, when they are set on your site you are responsible for them). Instead, click on the little "i" in the top left corner of your browser. There you should find the Twitter cookie.
There is no any filter, just delete all cookies form the tweeter domain and refresh wpForo how any time you want, you'll see it'll always be empty, the tweeter cookies come from tweeter website when you got to there. So they have no any relation to wpForo.
This cookie appears as soon as I enable Twitter sharing in the wpForo settings.
Yes, but click on that and see what's in it. It should be empty if you cleared all cookies before checking. The domain doesn't matter, if it doesn't have content then the cookie is empty.
@robert So what exactly is this if it's not a cookie? Because it still only appears when I activate the Twitter sharing option.
This is just nothing. This is just an empty space created to test by Twitter. For more info please contact to Tweeter support team. This is not created by wpForo for sure, and this doesn't content any cookie for sure.