Germans are recognized by humans and they don't have to attend US schools or pay US taxes
That's because Germans don't live in the US. If a German resided in the US, they would certainly be expected to do those things. These chimps will live in the US, so US laws would apply to them.
Germans were a bad example, corporations have legal personhood and they don't have to go to school, or pay taxes in a lot of cases but that's another debate entirely :)
But also relevant is people who are legally incompetent. Legally incompetent people don't have the ability to enter contracts or do any of the things you're worried about, they have a legal guardian who controls their affairs.
If the suit succeeded and they were declared legal persons I suspect the change would be the current owners of Chimpanzees would turn from owners into legal guardians. The main difference is instead of the Chimps being property to use as they wished (as long as they weren't abused) the responsibilities of that guardian would now be to ensure the welfare of the Chimpanzees.
I don't know the exact outcome they want
I'm thinking this point should be explored a little more deeply.
True but the fact it isn't really in the article doesn't mean they haven't thought it out. Sooner or later we're going to have to deal with animal welfare in a serious way, I find it really hard to believe that future generations won't look back at our current farming practices with extreme disapproval. As for our treatment of cognitively advanced animals like primates, elephants, dolphins, and possibly pigs, I suspect they'll also look down on that with strong disapproval.