There are many varieties of Kalman filter (KF), as more complicated algorithms were gradually developed to describe more complicated processes or to avoid certain kinds of numerical errors. There are some mathematical properties all of them possess, but there are other properties which apply to just one. Considering only the linear version (LKF) may lead you to draw conclusions which are incorrect for extended (EKF), unscented (UKF), or other kinds of related filters.
In almost all cases, the order of measurement processing does indeed impact the results, as physically it must. The essence of a Kalman filter is a model of how the state changes with time all by itself. The gain formula is a rule for updating your estimate of the state based on measurements, but the state would continue to change even if you stopped trying to measure it. Insensitivity to measurement order, or equivalence to batch least squares, is a property only of the linear Kalman filter with zero process noise and state transition matrix the identity. That would usually be a terrible idea, but there is one situation where it might come in handy.
The NASA book (150 pages!) you linked is all about filtering as a means of incorporating data into ephemeris estimates, so when they say "state", they mean the position and velocity of some orbiting object, plus any additional information of interest. Section 3.2 concerns a subtle aspect of incorporating variable measurement rates in a discrete time Kalman filter. If there is sufficient time between each measurement and the next, there is no need to do what they describe, and the order dependence of the resulting computation is desirable and physically realistic.
If, however, the time interval between some measurements becomes exactly zero, then things can go wrong. In that case only, the set of measurements which all pertain to a single time step should be treated equally, and there is no need for state updates between them, because in zero time there is zero change to the state (the object does not move). The algorithm they give is simply a way to avoid having to write a separate batch least squares estimator within each discrete time step by fooling your Kalman filter into doing the same job. It's not the most straightforward way to approach the problem, but since you're already using a Kalman filter for something else, you might as well adapt it to do that, too.