This idea seems a bit on the scale of a phenomenon, Bishop Fellay has authority because the Society perceives that he has that authority.
Look at the facts from the last year-and-a-half. Mgr. Fellay and Mgr Williamson had a very public dispute over the direction that the FSSPX ought to take. Mgr Fellay declared Mgr Williamson removed from the FSSPX. Mgr Williamson called for Mgr Fellay's removal as Superior General.
A year-and-a-half later, Mgr Williamson is part of the Resistance and not the official FSSPX. In contrast, Mgr. Fellay is still Superior General of the official FSSPX.
One can make all sorts of canonical arguments about whether or not Mgr. Fellay is legitimately head of the FSSPX, or whether his actions as Superior General are valid. But at the end of the day, they are being enforced by the majority within the FSSPX.
Apparently someone recognizes their situation as canonical, otherwise how do they perceive that he has authority.
Exactly.
Moreover, there is no higher human authority for Mgr Fellay's opponents to appeal to, that would be accepted by the majority within the FSSPX. One could appeal to God, who most certainly will be the judge come the afterlife, but so far in this lifetime God is allowing Mgr Fellay and his supporters to continue in charge of the FSSPX.
Which beggars the question, Is the SSPX giving itself its own authority?
I am sure if one were to ask Mgr Fellay and his supporters to justify their authority concerning these actions, they would appeal to "supplied jurisdiction" and "state-of-necessity."