The OP is confusing board certification with licensure.
BTW, the "AMA" neither accredits residencies nor grants licensure to physicians.
When you complete an ACGME-accredited residency (often mistakenly referred to as an "AMA" residency or an "MD" residency) you are eligible to take their board examination. You then become an MD/DO/IMG who is board certified by that specialty board (the ABIM for internal medicine, the ABS for Surgery, etc.)
When you complete an AOA-accredited residency, you become eligible to take their board exam and, if you pass, become certified by the appropriate osteopathic specialty board (the ABOIM for internal medicine, for example).
Licensure is a separate issue. In most states one can become licensed as a physician after one year of post-graduate training. The licensing is done by the state medical board. Usually there is only one board that oversees MDs and DOs but a few states have separate osteopathic licensing boards.
