“Kalman filter” usually refers to “linear quadratic estimator”, which assumes a linear model in its derivation. This will impact the “predict“ step at the very least, and I think also the way the uncertainty propagates. There are nonlinear estimators as well, though they usually have less-nice guarantees (eg particle filter, extended kalman filter)
Edit: in fact, I see part three of the book in tfa is devoted to nonlinear Kalman filters. I suspect some of the crowd (myself included) just assumed we were talking about linear Kalman filters