I am taught that there is no magnetic force on a moving charge if the
charge moves along the direction of the magnetic field. That would
mean that the magnitude of the magnetic field at that point would be
zero.
Careful: the second statement does not follow from the first. If we adopt a rectangular system of coordinates (e.g. the standard $x,y,z$ coordinates) then the formula for the force on a point charge $q$ in a magnetic field $\bf B$ is
$$
\begin{array}{rcl}
f_x &=& q (v_y B_z - v_z B_y) \\
f_y &=& q (v_z B_x - v_x B_z) \\
f_z &=& q (v_x B_y - v_y B_x)
\end{array}
$$
(This can also be written ${\bf f} = q {\bf v} \times {\bf B}$ but I want this answer to be accessible to people who don't know that notation). For example, if the velocity $\bf v$ is in the $x$ direction and the field is in the $y$ direction then these equations give
$$
\begin{array}{rcl}
f_x &=& q (0 0 - 0 0) = 0 \\
f_y &=& q (0 0 - 0 0) = 0\\
f_z &=& q (v B - 0 0) = q v B
\end{array}
$$
so in this case the force is in the $z$ direction with a magnitude $q v B$. So far so good.
Now let's see what happens when the velocity and the magnetic field are both in the same direction, say the $x$ direction. Now we find
$$
\begin{array}{rcl}
f_x &=& q (0 0 - 0 0) = 0 \\
f_y &=& q (0 B - v 0) = 0\\
f_z &=& q (v 0 - 0 B) = 0
\end{array}
$$
So now the force is zero but the magnetic field is not. So a zero force does not imply a zero field. What would imply a zero field is if the force came out zero no matter what direction the particle was travelling in.
Finally, in the case of a zero field you do indeed have a situation where it is not possible to assign a direction to the field. As a mathematical statement, that is true. But in physics it is hard to get a truly zero field. One can obtain a field whose magnitude is zero at some particular point, but near that point it would typically be not quite exactly zero (because it is so hard to get a strictly uniform field). In this case what happens is that one can assign a direction to the field at any location near to but not exactly at the special point.
