Some of my answer will referencing the following answer to avoid too much repitition: https://physics.stackexchange.com/a/452325/59023
That is, are the motional electric field and the magnetic field of the plasma really the same field, or are they two different fields that counteract each other and allow the plasma to keep flowing straight with velocity $u$?
The motional or convective electric field is just a Lorentz transformation.
Separately, a foundational idea in plasma physics is that if you have a plasma with high conductivity, the magnetic field is frozen in to the plasma, due to ohm's law...
It's not really Ohm's law so much as flux conservation in the limit of infinite conductivity. If the conductivity goes to infinity, the integrand of the time variation of the flux goes to zero (e.g., see https://physics.stackexchange.com/a/452325/59023 and https://en.wikipedia.org/wiki/Flux_tube).
Does anyone have an intuitive explanation for why $\mathbf{E} + \mathbf{u} \times \mathbf{B} = 0$ implies that the magnetic field moves with the plasma?
First magnetic fields don't move. Sources move and the field responds (well, QFT folks might object to this chicken and egg order but it will make sense shortly). The frozen-in condition is often inappropriately described in terms of moving magnetic field lines. You should not believe statements like this for two reasons. First, field lines are a mathematical tool we use to visualize vector fields, not a physical phenomena. Second, field lines don't move as they only have meaning for any given instant in time. That is, you must "re-draw" them for every instance and where they are drawn/defined/shown is entirely up to the user.
Try not to think of it as implying the magnetic field moves but more so that magnetic flux is conserved along a flux tube -- a cylindrically symmetric, locally, surface that maintains a constant flux at any given cross-sectional slice. That is, if you choose an arbitrary circular cross-section at some time and position then follow it as the system evolves, the cross-sectional area will change to maintain a constant magnetic flux through it. It just so happens that in the limit that $\mathbf{E} + \mathbf{u} \times \mathbf{B} = 0$, there is plasma flowing orthogonal to the magnetic field, which requires an electric field. So if you draw your initial circular cross-section, it will lie in the plane of the orthogonal flow and its outward normal will be aligned with the magnetic field. If the flux must be maintained, then this tube we initially defined by our arbitrarily chosen cross-section must be convected with the plasma flow. If it were not, then the magnetic flux would not be conserved.
Why is this "frozen in flux" condition the same (except for a minus sign) as the motional electric field? Is this just a mathematical coincidence, or is there a deeper relation between the two that I'm not grasping?
Well, the convective electric field arises because of the frozen-in condition through Ohm's law. That is, to maintain a constant magnetic flux, one requires there be no resistive dissipation, i.e., infinite conductivity. So in the normal, resistive world one would normally say the electric field is approximated by something like $\mathbf{E} = \eta \ \mathbf{j}$, where $\eta$ is the scalar resistivity and $\mathbf{j}$ is the vector current density (technically $\eta$ should be a rank-2 tensor for general systems). In an ideal MHD system, $\eta \rightarrow 0$ such that in the plasma rest frame $\mathbf{E} = 0$. Thus, if one does a low velocity, $\mathbf{v}$, Lorentz transformation, the electric field in stationary lab frame is now given by $\mathbf{E}' + \mathbf{v} \times \mathbf{B}' = 0$. The Lorentz transformed magnetic field in the non-relativistic limit is unchanged so we have the frozen-in condition.
