Why is the force of a magnetic field or a magnetic field not a vector field?

The magnetic force is not a vector field because vector fields are functions $\mathbf F:\mathbf r \mapsto \mathbf F(\mathbf r)$ that take a single vector value at each position, and the magnetic force $\mathbf F=q\mathbf v\times \mathbf B(\mathbf r)$ also depends on the velocity of the particle that's experiencing the force.

You could ask, instead, for the magnetic force on a particle with a given velocity, which itself may or may not depend on the position, i.e. $\mathbf v=\mathbf v(\mathbf r)$ (where that dependence may just be a constant), in which case you'll get a map $$ \mathbf F:\mathbf r\mapsto \mathbf F(\mathbf r)=q\mathbf v(\mathbf r)\times \mathbf B(\mathbf r) $$ that does have a unique value at each position and which therefore does define a vector field, of which you can ask e.g. whether it's conservative or not. However, until you actually define what velocity dependence $\mathbf v(\mathbf r)$ you want to take, none of those terms are applicable.

It is indeed a vector field in the sense that it associates a vector to every point in the space, except that the vector is in $T_p^*M\otimes T_pM$ rather than $T_pM$, where commonly used vectors live.