I've thought about this before, in addition to why Black people love Spongebob Squarepants. I have a few points.
1. Like most people said here, most millennials grew up with this stuff (myself included), Black people are far from exclusive. I've definitely seen plenty of white guys still sperg about DBZ.
2. DBZ was cool at the time because most other cartoons had "kid gloves". Most American action cartoons rarely showed people beating each other up and death. Black kids are particularly drawn to that, they don't want to be talked down to (as did most white kids). Spongebob Squarepants was similar in a way because it had no morals, no "very special episodes". While most post-Ren & Stimpy cartoons adopted its gross-out humor and comedic style, only a few maintained its irreverence, Spongebob being the most obvious. Ultimately, it comes down to the fact that Black kids probably saw Western cartoons with morals and overt sentimentality as "white authority figures preaching to you", which is of course anathema to black people. DBZ was about kicking ass, Spongebob was about having fun, period.
3. Probably the most simple explanation is that anime isn't made by white people. This one also ties into why Blacks of an earlier generation loved Kung-Fu movies. Pretty much everything that came out during that time was made by white people, so when they see someone "non-white" kicking ass, they love it. It doesn't matter that there is very little solidarity between the Asian and Black communities (as a few armed grocery store owners might show) or that native Asians don't give a shit about racial politics, all that matters is that Black people had someone to look up to that wasn't white. Bruce Lee and Goku were far more appealing than Warren Beatty and Batman.
4. Masculinity. This ties into point 3, Black men are hyper-masculine and love fighting, and the heroes in shonen anime, kung-fu movies, and fighting video games are a good outlet for that if they can't box. Also, most black men are fatherless and those non-white masculine figures can fill in the gap.
5. As many people have noticed here, most Black people are just into shonen anime that was shown on Cartoon Network. However, I have noticed that when a Black guy gets into some real weeb shit, they get into waifu culture hard. I think that partly has to do with the fact that anime is much more overtly sexual, with even non-hentai series having plenty of fanservice, which appeals to hyper-sexualized Black culture. Compare that to most Western media, where sexuality tends to much more toned down outside of pornography (partly because of religion in the past, now because of feminism). The Japanese aren't afraid to show off the boom anime babes that make you think about the wrong thing.
6. Reality sucks right now, doubly so for millennials, triply so for Black people. When reality sucks you tend to regress to your happy place, which is childhood.