In the book of boundary layer theory:
"The solutions of the Navier–Stokes equations do not have to be unique for given initial and boundary conditions. Primarily because of the nonlinearity of the differential equations, variation of geometric or fluid mechanical parameters can lead to bifurcations in the solution and thus to multiple solutions."
What exactly dose geometric or fluid mechanical parameters mean?
When it comes to partial differential equations in physics, non-uniqueness of solutions seems to be the norm, rather than the exception. Another example would be in electrodynamics, where you get unphysical solutions to the Maxwell equations by using the advanced Green's functions, rather than the retarded ones.
In the case of the Navier-Stokes equation, this problem is worse because the equations are non-linear. In fact, the question of whether the Navier-Stokes equation always has "nice" solutions is still unsolved, and one of the millenium problems with a million dollar prize on it.
If you look at the derivation of Navier-Stokes, there is no real reason to expect any different. One usually proceeds by taking a volume of fluid and applying Newton's laws. By taking a limiting procedure one then obtains a partial differential equation. All this proves is that fluids obey the Navier-Stokes equation. Nowhere in this kind of argument does it say that every solution has to be physically relevant.
Regarding the geometric parameters you mention, the Reynolds number is one of them, although you can construct others, depending on the relevant physical scales in the system.
Geometric parameters can be length scales in your problem (radius of inlet, width of wall), fluid mechanical parameters I think might be material parameters (such as equations of state) and reaction rates, stochiometric equations and such, typical velocities of the fluid, temperature, ...
You need to combine these into dimensionless numbers to create an analogous problem that is cheaper to test in. So when all the parameters like Reynolds number and so on are the same you will get similar solutions.
Bifurcation is a term used for example in flow charts for ODE solutions in chaos theory. When it can go in different trajectories (even for small perturbations).
Regarding if there are multiple solutions given same data, maybe it is meant there is a gauge theory? But not sure.
