SharonA, my maternal grandmother came to the USA from Barbados and my maternal grandfather came to the USA from Trinidad.

My biological father was adopted. I believe his family was from New York state. I know that he was lighter skinned than my mother {whose mother's mother was White}. But I don't know anything about my father's ancestry. I can presume that my father had some White ancestry only because of his appearance.

I believe that I don't have any Southern Black ancestry-or any other ancestry that I know of-from the American South. It's possible that I have some ancestors who were enslaved in the American North or the South {from my father's side of the family}, but again, I don't know that for sure.

All this to say, I believe that all persons who are African Americans don't have to be descended from people who were enslaved.
And I believe that just like other words and phrases, "African American" doesn't have to make perfect, literal sense.

Imo, "African American" is a cultural referent, much more than it is a geographical and/or biological referent.

Also, SharonA, here's my comments regarding your question about whether your relative who is of mixed racial ancestry "just get stamped as "black" or "Black American" without taking the other half of his heritage into consideration?" :

On a number of threads including the one whose link I provided, I wrote that I consider the "one drop of Black blood" rule to be racist. I hope your cousin's son learns about and celebrates all of his heritage.

In the USA, people have a certain degree of free will about what race/ethnicity they say that they are [in the census and on applications and forms]. However, a person's physical appearance will largely determine whether people in the real world accept his or her racial/ethnic self-identification. If a person is of "ambiguous race or ethnicity} {meaning other people can't readily tell which race or ethnic group that person "belongs to"}, then that person may have some difficult experiences in everyday life [then again the person may not have such difficulties depending where he or she lives. I seems to me that the more multi-cultural and multi-racial his or her community, the better and easier it would be for a person of ambiguous ethnicity.

With regard to who is or is not African American:
I know a number of people who have one birth parent from the continent of Africa and one birth parent who is African American. These people consider themselves to be African American [and are considered African American by other folks including those who are African American].

I've known Black people with two Black birth parents who could easily pass for White. I've known Black people who have one Black birth parent and one White birth parent who are darker than me [and I have two Black birth parents]. I've also known people who physically appear to be African Americans who identify themselves as Native Americans because that was the primary culture they were raised in.

I also know some people of African American/Filipino ancestry. These people consider themselves to be African American, probably because they live in the United States and they understand that most African Americans are a mixed race people.

And I could cite other examples.

I also understand that people from North African nations such as Ethiopia, Egypt, and the Sudan are considered to be White by the United States regardless of the color of their skin or their other physical features. They may look "African American" but legally they are White in this country.

And I'm aware that non-Black people {for example, White people and East Indians} who were born in Africa or who live in Africa and now live in the USA could also call themselves "African Americans". But that is a different than the standard meaning of that referent in the USA.

I believe that with regard to racial referents, particularly mixed racial referents and referents like "half-African Americans" times, they are a-changing. This may be a good thing, and it may not be all that good. But it will be what it will be.