> or do you actually have to hunt for your own projects, like you might in some consulting firms?

This is how the company works on a fundamental level.

On healthy teams, having something assigned to you (for levels under staff/6) is normal. On unhealthy teams, you're just a sitting duck and it's better to find your own work. Or else you'll be forced to work on bullshit projects with no upside.

Side note: the "they" who does the assigning is not a manager, it's another IC. The ones that go out and find their own work. That could be at any level technically, but usually staff+ because they form little political mafias.