In addition to brand name, there are specific Sith teachings and knowledge.
All the Sith orders can trace their teaching lineage back to the Hundred Years Darkness and the Sith that came from there in one way or another. Whether direct teacher-student lines, or in a few cases studying at the feet of holocrons of past Sith masters.
Sort of like how a random female darksider could claim to be a Nightsister, but unless she learned their style of magic, she's really not one and is just using the name.
Ventress claimed to be a Sith, but she had no training from anyone with Sith history at the time, she just combined Nightsister and Jedi training. Once she had some training with Dooku, she could more legitimately claim the title as she now had proper Sith teachings (if not as much as a Sith Lord Apprentice), and if Sidious and Dooku keeled over death, I'm sure she would've.
Which is pretty much what Lumiya did. I don't have a lot of respect for Lumiya's sith on the whole, but she was trained by Darth Vader in the ways of the Sith.