Can a person over 65 be on both Medicare and Medicaid Insurance?

Yes, a person can be on both; that is call "dual eligible". To qualify for Medicaid you must be within the income and asset limits that your state sets. You won't give up your SS in most cases. If you are in a nursing home or long term care center and Medicaid pays for the home you'll then give up most of your SS.

If you qualify, Medicaid can pay your Part B premium and in some states they'll also pay some if not all of your Part D premium. You'll also get your medications at a reduced rate in many states. Depending on your state and your eligibility level Medicaid will pay whatever Medicare doesn't. There are also Medicare Advantage plans specifically designed for dual eligibles in many areas.