Itchy skin on various places. does cat allergy or dog allergy do this?

Yes: Contact with cat or dog dander can cause redness and itching or even hives.

With direct contact: Cat and dog allergy causes eye, nose and asthma symptoms and these symptoms usually develop before itchy skin. Direct contact with your pet is required to cause itchy skin and then you would see redness and possibly hives at the contact site. If your pet spends a lot of time on your bed or sofa and then you lie on it, you could develop itchy skin.