Anthem and Blue Cross Blue Shield are the same company however the difference between the two are that Anthem Blue Cross Insurance is a for profit company.
The Company Blue Cross Blue Shield Insurance is a non profit company but they're owned by the same company so technically they are the same insurance company but they both work differently when providing insurance and health insurance coverage to people.
So to get the correct health insurance for you then you need to use either Blue Cross Blue Shield or Anthem Insurance depending on how you need your insurance set up.