How is it passed paternally? From my knowledge, the "sayedness"(?) was passed from Lady Fatimah AS not from Amir ul-Mu’mineen AS. Like, from my knowledge, al-’Abbas Abu Fadhl AS is not a sayed, even though his father is Imam ’Ali AS.

Is there a reason for that?

Abul Fadhl was a Sayed no doubt, and the laws of Sayed (like Khums applied to him). Khums laws apply to paternal Sayeds not maternal ones.