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I don’t know what PhD program your brother was looking at but ideally candidacy committees won’t pass you unless you chose the problem or made it your own. Where your brother is right is that forming such a ideal committee is mostly politics.

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I don't know about math, but in CS, a lot of funding comes from specific grants, so you'll be working on something related to what your advisor got grant funding for.

Regardless of grants, etc., your general area of work is going to be based on what the faculty members' focuses are. But in any case, you'll probably still be doing your own sub-piece of some larger effort. You wouldn't choose an advisor at random, so this isn't really relevant.

I mean, if you're interested in astrophysics, you shouldn't go to a school where the Physics department is focused on nuclear physics!

I think you are right on both points. There were more details and his answer was just a quick way to say:

He didn't care about the status of having a PhD

He didn't need a PhD to advance his career

He was having fun doing math stuff anyway

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