Generally, if a person Indicates that they are Irish-American, is that reference to heritage, or simply the place they were born?
See? That wasn't so hard was it?
I'd say it's most often identifying a heritage, though most of what I posted about in terms of using it legally, on forms. But okay, let's talk about in terms of heritage.
Something like Irish-American is generally not used to identify race, as much as heritage and culture. When Blacks use "African-American" it is much more about racial identity. A Black family which grew up generations in Italy, migrating to the US could prefer to identify with "Italian-American", much easier than the reciprocal, a White family in Africa identifying as African-American. You've said as much. In the one case, it's just a term indicating where their culture and heritage came from without as much regard to race. In the case of African-Americans, that is socially reserved for the race, irrespective of culture or heritage. So obviously the two aren't really playing by the same rules.
So a question for you. The culture of people who tend to identify as African-Americans, how much of that culture is actually African culture? How much of it evolved from slaves which developed their own unique culture along with adapting some cultural features from Southern Americans? What similarities in culture do Black Americans share with Africans? And, since "African" isn't anything close to homogeneous. Which Africa? Muslim? Christian? Tribal? Westernized?
I kinda suspect that "African-American" is a label which derives "Africa" from their ancestral origin, but their culture isn't as African as it is unique to American Blacks. And if that's true, what is wrong with saying so?
I think the term attempts to find an identity that Black people can be proud of, that makes Black Americans distinct from other cultures associated with Whites. And there's nothing wrong that. Nothing wrong with holding the familial cultural traditions. It's the race part that gives me pause. It just makes the "African" part of the label, maybe not the best fit. The people who identify as, say, Irish-Americans aren't in the same boat, because they'll likely drop that hyphenation after living here a few generations. But the extent to which race is a part of the identity of African-Americans, that identity term just doesn't fade away over generations.