The ingredient is talc itself. Quoting from cancer.org:

Many studies in women have looked at the possible link between talcum powder and cancer of the ovary. Findings have been mixed, with some studies reporting a slightly increased risk and some reporting no increase. Many case-control studies have found a small increase in risk. But these types of studies can be biased because they often rely on a person’s memory of talc use many years earlier. Two prospective cohort studies, which would not have the same type of potential bias, have not found an increased risk.

Furthermore, old talc (before 1975 or so) used to be contaminated with asbestos. From the same source:

IARC classifies talc that contains asbestos as “carcinogenic to humans.” Based on the lack of data from human studies and on limited data in lab animal studies, IARC classifies inhaled talc not containing asbestos as “not classifiable as to carcinogenicity in humans.” Based on limited evidence from human studies of a link to ovarian cancer, IARC classifies the perineal (genital) use of talc-based body powder as “possibly carcinogenic to humans.”

Regarding J&J, DavePhD is correct that there was a 1964 J&J memo which said that talc was not "absorbed safely" by the vagina (this was in the context of J&J looking into alternative products). Furthermore another 1975 J&J memo talked about "Talc and the ovaries" in which a 3rd party study about talc and ovarian cancer was being discussed. Both of these memos were entered into evidence at the trial.

Another 1986 J&J memo entered into evidence acknowledged that numerous "studies have implicated talc use in the vaginal area with the incidence of ovarian cancer". A 1992 J&J obstacles/opportunities file listed as a "major obstacle", the "[n]egative publicity from the health community on talc (inhalation, dust, negative doctor endorsements, cancer linkage". A 1994 talking points summary for a company executive contained an acknowledgement that some scientists reported a link between "use of talcum powder and ovarian cancer, however that same document denied that "talc is not safe".

Starting in 2006, the updstream provider of talc for J&J (Imerys) updated its MSDS (included with every shippent) to include the IARC warning about that "perineal use of talc-based baby powder is possibly cancerinogenic to humans (Group 2B)." Yet J&J never included this information in materials for its consumers. The woman's attorney wrote (in opposition to JNOV):

From this, as well as other evidence, the jury could have reasonably concluded that Johnson & Johnson should have provided a warning and its conduct was sufficient for punitive damages" because prior case law had established that ignoring MSDS warnings was "'despicable conduct' with wilful and conscious disregard of the safety of consumers. Cal. Civ. Code 3294(c)"!

IANAL, but methinks any other company that received similar talc MSDS and didn't update their consumer info is quite liable to get sued in California. As of Sep 30 (the lastest news I found), the judge had yet to issue his ruling on J&J's JNOV though.

The same lawsuit also named Imerys as a defendant, but the judge dismissed that claim before trial. Reuters says that one of J&J suppliers also lost one of the many talc lawsuit, but it doesn't name the supplier (I'm guessing its Imerys though). A small company that did the packaging for J&J's talc product and followed their instructions regarding labelling has also been sued. I'm not [yet] aware of any other big company having been sued or having lost a talc lawsuit, but there are thousands such lawsuits all over US presently...