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Yes, frequentist inference only allows you to reject the null hypothesis (cats do not cause mental illness), but not to accept it if you fail to reject [1].

The abstract as written is simply wrong from a statistical point of view:

our study strongly indicates that cat ownership in pregnancy or early childhood does not confer an increased risk of later adolescent PEs [2]

If you instead adopt Bayesian reasoning, then the direction of the measured effect is weak evidence that cats cause illness later in life. Either way you cut it, the study author's interpretation fails to coincide with the actual data they collected.

[1] https://liesandstats.wordpress.com/2008/09/08/accept-the-nul...

[2] https://www.cambridge.org/core/journals/psychological-medici...