Trouble is, Clans in Vampire the Masquerade aren't organizations, but bloodlines - meaning they're specific types of vampire, each with its own unique biological quirks and even specific superpowers.
A member of the Tzimisce clan will always be one of the Tzimisce bloodline, able to trace their lineage back to the Eldest, the Tzimisce Antediluvian. You can't have a Toreador vampire become a member of the Tzimisce clan, because they weren't Embraced into the Tzimisce clan - they were Embraced into the Toreador clan (and vice versa).
They will always possess the clan-unique discipline of Vicissitude, they will always need to sleep in their native soil, and they will always be targeted for assimilation by the Eldest when Gehenna finally dawns. No exceptions, no variations except for the Caitiff and Thin-Bloods brought about through excessive breeding. Ergo, species rather than organization.
Clans themselves are often too decentralized to act as organizations anyway: because vampires are so often at each other's throats, getting them together is a tricky business except as a Sect - with the exceptions of the Tremere Clan, who are all blood-bound to serve as a unified front, and the Nosferatu Clan, who have to band together in order to avoid being hunted down by the Nikutu. And even in those cases, you still end up with traitors like the Antitribu - all of whom are still counted as members of the Clan despite being traitors to the Camarilla.
Finally, each Clan as a whole generally serve as the membership of an actual organization, namely a sect:
The Tzimisce are almost always members of the Sabbat, alongside the Lasombra clan, the Antitribu of various other clans, and many others.