Can thrombocytopenia be caused by a cat (some disease from cats)?

Probably not. Cats are not likely to do this, unless there is a very extreme allergy that triggers a severe autoimmune reaction, and this is real remote. Cats usually spread toxoplasmosis infection as they pass it in their stool, but this is controlled by the immune system. A cause of low platelets can be snake bite like from a pit viper.