Can living with pets cause microcytic hypochromic anemia?

No: Microcytic hypochromic anemia is usually due to iron deficiency, and should be unrelated to living with pets.

I seriously doubt it: The most common reason for microcytic anemia in the us, overwhelmingly, is an fe deficiency state. This is easily tested for and treated. Frankly, because the tests are more costly than the treatment, and the odds are in favor of fe deficiency, most providers will simply try a course of fe and check impact. If fe deficiency, its important to figure out cause and correct it otherwise anemia recurs.