How to relate magnetic flux to fluid flux? I'm trying to grasp magnetic flux as an analogous concept to fluid flux, which is more concrete in my opinion. For example, we can define a vector field $\mathbf{F}(x, y, z)$ which describes the movement per unit time of a liquid's molecules at a certain position in space. Then, we can measure the volume of liquid flowing (total movement) through a surface $S$ per unit time by taking the flux as so:
$$\iint_S \mathbf{F} \cdot \mathrm{d}\mathbf{A}$$
But when I encounter a magnetic field $\mathbf{B}$ that describes a quantity with units $\textrm{Wb}/\textrm{m}^2$ what does this intuitively mean? Is there an equivalent intuition of charged particles moving through a surface from a force caused by the magnetic field?
 A: Intuitively. 
Fluid flux tells out how much fluid goes through the surface element (technically perpendicularly to it).
Magnetic/Electric fields' fluxes tell your how many field lines go through the surface element. If you think of $\mathbf{E}$ and $\mathbf{B}$ fields as "fluids" that permeate space, then it's the same thing: the flux measures how much of these fluids cross your specific area.
The fluxes are defined independently of (test) charges and currents. They are properties of the fields themselves (which, sure, are generate by some [source] charges and currents).
A: In my personal opinion, grasping for a more physically intuitive analogy in this particular instance likely does more harm than good.
As you know, flux is a concept which arises when you consider a surface $S$ in the presence of some vector field $\mathbf v(x,y,z)$.  If you divide the surface up into infinitesimal little pieces with area vectors $d\mathbf A$ (where the direction points normal to the plane of the little piece), take the dot product $\mathbf v \cdot d\mathbf A$, and add up the contribution for every piece, we call the result the flux of $\mathbf v$ through the surface $S$.
Sometimes this quantity has a clear, physical meaning.  When $\mathbf v$ corresponds to the velocity of some flow (as in the case of the velocity field of some fluid), then the flux through a surface simply gives the rate of fluid flow through the surface (in, say, $m^3$/$s$).  In other situations there is no such obvious interpretation, and such is the case with electric and magnetic flux.
If you visualize electric and magnetic fields by using field lines, then the flux through a surface correlates to the number of field lines which penetrate the surface (where positive flux corresponds to the field lines penetrating parallel to the surface normal).  I would not say that the flux is defined to be the number of field lines which penetrate the surface, because as you say, that is not a well-defined quantity.
Trying to shoehorn flux into a more intuitive box than that is probably not a good idea.  It might wrongly lead you to think that flux is a measure of the dynamic flow of some quantity or material (it isn't), or that it implies that there is some kind of physical motion taking place (there's not).  It's just a mathematical quantity which you can associate to a surface in a vector field, no more physical than taking the derivative of a function.

That being said, Maxwell's equations tell us how electric and magnetic flux through surfaces relates to physical things like the total electric charge enclosed, the total current penetrating the surface, etc.  These things are, of course, extremely physical, so building intuition based on these things is a very good idea. 
A: you’re problem is that you are thinking of “particles moving through a surface in a unit time” as the fundamental definition of “flux” when in fact particles moving through a surface in a unit time is only a particular kind of flux. 
Generally flux is the integral of a vector field over a surface. In the case of mass/charge/fluid flowing past a surface the vector field is the velocity multiplied by the mass/charge/density. The mass or charge current etc. that is, for a fluid the mass flux is the integral of the mass current (vector field) over the relevant surface. 
For electric or magnetic flux we drop any requirements about something moving but the concept of integration over a vector field remains. 
