The Dalmatic is not related to the Sakkos, the Sakkos is strictly an Imperial garment that only the Emperor could wear. When there was no longer an Emperor the Patriarch of Constantinople started to wear the Sakkos and the Crown and soon there after it spread amongst all the Bishops.
Of course, they aren't historically related or identical, but they are analogous.
Further, the sakkos was used by many Patriarchs and Archbishops on major feast days, as attested as early as the 12th century by Balsamon -- well before the Fall of Constantinople. It became ubiquitous later on.
Really, all of the eastern vestments being discussed were Imperial garments at one time or another. If you go back far enough (fourth century), even the Dalmatikon was for the Roman Emperor, then only for high-ranking Imperial officials, then for some high-ranking hierarchs, and so on. In that sense, there's really nothing all that unique about the sakkos being an Emperor-only garment.