California has had a trademark law on its books since 1941.  1941 Cal. Stats. Ch. 58.  The legislature repealed that law in 1967 and replaced it with the Model State Trademark Law drafted by the International Trademark Association ("INTA").  1967 Cal. Stats. Ch. 1556.  In 2007, the California legislature repealed the 1967 law and replaced it with the 1996 version of the Model State Trademark Law drafted by the INTA (AB 1484 (Krekorian), 2007 Cal. Stats. Ch. 711).  The current law is codified at California Business and Professions Code § 14200 et seq.  Free PDF images of trademarks registered under the law are available on the California Secretary of State's website.

"It's a scandal.  It's an outrage."

Despite the law's long history, it may nonetheless be unconstitutional (at least in part) as a result of recent decisions by the U.S. Supreme Court.  Two years ago, the Court declared unconstitutional the federal trademark act's proscription on registration of marks that "disparage" any person living or dead.  Matal v. Tam, 137 S. Ct. 1744 (2017).  Notwithstanding this decision, California's statute continues a similar ban on disparaging marks.  Bus. & Prof. Code § 14205(b).  Yesterday, the Supreme Court, in an opinion by Justice Kagan, found the federal trademark act's ban on "immoral or scandalous" marks to violate the First Amendment to the U.S. Constitution:

"There are a great many immoral and scandalous ideas in the world (even more than there are swearwords), and the Lanham Act [the federal trademark act] covers them all.  It therefore violates the First Amendment.

Iancu v. Brunetti, 2019 U.S. LEXIS 4201.

California's law also bans marks that are immoral or scandalous.  Bus. & Prof. Code § 14205(a).  Thus, Iancu almost certainly invalidates California's ban as well. 

Incidentally, this is a subject that I first tackled four years ago.  See Can The Secretary Of State Reject Disparaging Business Entity Names?