Despite the inherent problems with trying to prove a negative, I feel comfortable calling BS on this one.
There is no (internet) evidence for this assertion when searching for "Soundgarden cult", "Chris Cornell cult", or even "Kim Thayil cult." Same is true if you replace "cult" with "religion" or "movement". There is no (internet) evidence for the Soundgarden/Cornell/Thayil "starting a religion", "starting a cult", or "inspiring a religion". Similarly, there is no evidence of "Black Hole Sun Cult" or "Black Hole Sun Inspiring Religion" or likewise. Limiting these searches to pre-2000 articles also shows no evidence.
By "no internet evidence", I mean anything that seems to corroborate this assertion outside of the trivia questions you mention. None of those sites provide any sources. The most information I could find mentions the same fact the original post does -- that it inspired the "Concerned Christians" cult. Even then, searching for Soundgarden or Black Hole Sun or Cornell or Tayil along with "Concerned Christians" still pulls up nothing remotely validating.
Furthermore, the claim that "Black Hole Sun" was the reason for Concerned Christians was founded is demonstrably untrue. Black Hole Sun was released in 1994 (album in May, video in June). Concerned Christians was founded in the 1980s (source, source, source)