Asthma triggers are personal, and cat-sensitized people react to allergen proteins riding on fine particles, with visible fur mainly a carrier rather than the agent.
Suitability should be decided by allergen testing and a pulmonologist; for non-sensitized asthmatics, the main risks remain their own triggers like dust mites and cold air.
Sensitized households that keep cats can enforce strict exposure control: purification, cat-free bedrooms, frequent textile cleaning, and physician-guided medication.
Research on childhood cat exposure and asthma is mixed, so neither causes nor prevents is a fair claim; individual medical decisions belong to doctors.