It's complicated.
Brave was founded by Brendan Eich, long-time Mozilla developer, long-time Mozilla CTO, briefly Mozilla CEO.
Brendan Eich also happened to be an anti-gay-marriage activist and donator (afaik, only his spare time, not at work) and California law says that if you donate more than [some sum of money], you must specify your employer. So, suddenly, a few days after Brendan Eich was nominated CEO, someone on Twitter published the records of Brendan Eich's donation, phrasing it as "Mozilla is donating against gay rights", or something like that. A shitstorm started, including death threats against Mozilla employees, a highly hypocritical PR campaign by OKCupid (whose CEO was member of the same political circles as Brendan Eich) to boycott Mozilla, etc. Brendan Eich was convinced to resign from Mozilla after about two weeks of this.
That's probably what GP calls "banned from Mozilla".
Of course, after this, US Conservatives rephrased this into "Mozilla is persecuting Christians", so the death threats continued, just from the other side.
Source: I was there.