Skin cancer can form after breast cancer, but I am not sure that is technically what happened in your mom's case. It sounds like she possibly had some radiation dermatitis after the original treatment, which is unfortunately common, but then possibly the original breast cancer itself spread to the skin. This is confusing because cancer is really identified by what cell it originates from, not necessarily where it is in the body.
I think it sounds like your mother had breast cancer cells IN the skin, rather than skin cancer itself, which is caused by cells that live in the skin and look different from breast cancer cells. Of course, without seeing it myself, I can not be sure.
Skin cancer (basal cell carcinoma or squamous cell carcinoma most likely) could possibly develop after damage to the skin from the radiation treatment, and lowering of her immune system by all this difficult chemo, but it usually would not occur right away, and would look different to the pathologist.
I see that your mom is having a tough time going through all this, and I am sure you are, too. Hang in there and remember you can always see a board-certified dermatologist for anything that develops with her skin or outer mucous membranes (mouth, genital area) to get some comfort while trying to get through the treatments.