Trying to create an analogy with common experiences seems useless; if I were running north through a west-flowing "field" of some sort, I wouldn't expect to suddenly go flying into the sky.
This is a reasonable expectation, since the electric and gravitational fields do make forces that are in the direction of the field. So let's try to see what goes wrong if we write down a force law for magnetism that behaves in the same way. The first thing we could try would be
$$ \textbf{F}=q\textbf{B} \qquad (1) $$
Well, this doesn't work, because such a force would behave in exactly the same way as the electric force, and it would therefore be the electric force, not a separate phenomenon. Magnetic forces are supposed to be interactions of moving charges with moving charges, so clearly we need to include $\textbf{v}$ on the right-hand-side. One way to do this would be the standard Lorentz force law, but we're looking for some alternative that is in the direction of the field. So we could write down this:
$$ \textbf{F}=q\textbf{B}|\textbf{v}| \qquad (2) $$
As an example of what's wrong with this one, suppose we have identical charges $q$ bound together with a spring. If they're sitting at rest in equilibrium, equation (2) says there's no magnetic force on them. But suppose we start them vibrating just a little bit. Now they're going to start shooting off in the direction of the magnetic field. This violates conservation of energy and momentum.
Fundamentally, this comes down to an algebraic issue. The vector cross product has the distributive property $(\textbf{v}_1+\textbf{v}_2)\times\textbf{B}=\textbf{v}_1\times\textbf{B}+\textbf{v}_2\times\textbf{B}$, and in the example of the charges on a spring, with the actual Lorentz force, this guarantees that the magnetic forces on the two charges cancel out. We really need this distributive property, and in fact it can be proved that the vector cross product is the only possible form of vector multiplication (up to a multiplicative constant) that produces a vector result, is rotationally invariant, is distributive, and commutes with scalar multiplication. (See my book http://www.lightandmatter.com/area1sn.html , appendix 2.)
The magnetic field created by the wire is azimutal while the field created by the solenoide can be studied as the sum of two contributions. The first one is an uniform longitudinal field inside the solenoid and the second is an azimutal field outside the solenoid that varies as I/r. Therefore the magnetic force the wire experiences is null because current and B are parallel or because the current does not see the field. Alternatively you can analize the magnetic force the solenoid experiences. The current on it has two components. The horizontal component does not sense force because it is parallel to B field. The vertical is attracted by the central wire, but after integrate the total force will be zero by symmetry considerations.
Best Answer
The universe is not preferentially selecting one direction over another. The fact that it appears that this is happening is an artifact of how we represent the magnetic field.
It is well-known that the existence of magnetic forces can be inferred from a Lorentz-invariant theory involving electric forces. For example, see this answer.
The magnetic force so derived necessarily has the property that parallel currents attract while antiparallel currents repel.
The magnetic field can be thought of as being the field that needs to be introduced into the theory in order to give a local description of this attraction between parallel currents. It is therefore necessary for the Lorentz force law to be written in such a way so that it gives the correct direction for the magnetic force between two currents. Otherwise the law would violate the observed Lorentz invariance of our universe. A law itself does not determine what actually happens; that can only be determined by experiment.
Because the direction of the magnetic field is assigned through a right-hand rule, a second application of the right-hand rule is needed in the Lorentz force law in order to get the correct direction for the actual force between the two currents. If the magnetic field direction were assigned through a left-hand rule, the Lorentz force law would also involve a left-hand rule. In neither case does the universe enforce an arbitrary choice of one over the other. We are simply describing the phenomenon in a way that requires us to put in the rule by hand in order to get the correct result.
This contrasts with the situation with weak interactions, which really do violate parity symmetry.