Two Different Sorts of Minorities
One division, within the category of people who are born into a minority group, is whether the parents are also members of that group.
Examples of groups where minority-group children are raised by parents belonging to the same group include ethnicity, religion (in fact the child's religion is largely caused by the parent's religion), and some genetically-based disabilities. Examples of groups of minority children raised by majority parents include almost all gay kids, most disabled kids, and a few kids of ethnic minorities (either adopted or mixed-race with a single parent).
The big difference is in the experience of community. The minority child who grows up with minority parents typically lives in two different communities. One is the majority community, where the power lies but also the source of discrimination and exclusion. But as a counterpoint to the majority community, they have their own community, which often lacks resources but provides a place where they can truly feel that they belong.
This has good points and bad points. The good points are the sense of belonging, as well as the ready availability of role models whose experiences truly resonate with the child. However, often the child feels caught between the two worlds, between the greater acceptance found in their own community and the greater opportunities in the other community. This can be seen, for example, in the issue of school success for poor (and often ethnic minority) kids in many large cities in US. If they work hard and try to be successful in school, that can mean turning their backs on their community.
The minority child with majority parents doesn't have the same experience of community. For them, unless their parents make have minority contacts, they belong only to the majority community. But they don't really belong, because this community discriminates against and excludes them. Even their parents may be discriminatory. And the only role models they have are different from them in an important way.
The good points are that these kids often have more opportunities, because their parents are more successful. This type of minority group is also much more likely to have majority-group allies, who advocate with and for them - almost all parents want what's best for their child. In the early years of the fight for rights, these allies may be the only ones who can get people to listen to them. Majority-group parents may be less likely to take certain forms of discrimination for granted (especially when this discrimination extends to themselves), because someone who is accustomed to respect readily notices when they're suddenly being disrespected.
But the bad side is a lot of loneliness. The children may feel like they don't really belong, even when among people who accept them, because the difference is still there. Self-made communities are a big help (and extremely common for such minorities) but can't completely erase the loneliness of growing up different from most or all of the people you know. In addition, it can be harder for these kids to find support, because there are some aspects of discrimination you can only really understand if you've lived through it. Lastly, adult role models may be very hard to find, leading the child to either identify with someone from the majority group, or have pretty much no idea what the future holds.