I feel like the title implies that cats cause schizophrenia when what's more likely is that people who have schizophrenia are more likely to like and own cats, and schizophrenia tends to run in families.
Cats can carry toxoplasmosis which seems to have a correlation to schizophrenia. This could be the cause of crazy cat lady syndrome: https://pubmed.ncbi.nlm.nih.gov/29068607/
Despite what a lot of people think, most pet cats don't have toxoplasmosis, and even those with toxoplasmosis rarely actually pass it on to humans, so that really shouldn't occur often enough to be measurable if it was transmission from cats causing it. You're more likely to get toxoplasmosis from uncooked meat, since that's how most humans are infected.
Now, people with toxoplasmosis are more likely to own cats, so there could be a correlation there, but at the same time, diagnosis for schizophrenia is more common in countries where there are less people infected with toxoplasmosis overall (I think in the US only like 11% of the population has been infected before). I can't say I feel like that would sufficiently explain the correlation.