Does access to Medicaid guarantee poor people the same quality medical care as those who are privately insured?
In general, it does not. A major reason for this is economic. With the way that Medicaid works, doctors have an incentive to refuse to accept Medicaid patients.
The government sets the prices that can be paid out to physicians who care for patients who are on Medicaid. Sometimes, the government of various states push the prices that they pay physicians down too far from the physicians' perspective. When this happens, some physicians will stop seeing Medicaid patients. At that point, ironically enough, a government program that is meant to guarantee access to high quality health care actually does the opposite.
Because of low payments to physicians, then, Medicaid does not always guarantee that poor people get quality health care.