Is the kernel the same as the union of all level sets (or perhaps the set of all level sets)?
Kernel (https://en.m.wikipedia.org/wiki/Kernel_(set_theory)): $\left\{\, \left\{\, w \in X \mid f(x)=f(w) \,\right\} \mid x \in X \,\right\}$
Level set (https://en.m.wikipedia.org/wiki/Level_set) $L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c \right\}~,$
Why is the kernel described as the vectors that are mapped to zero in vector spaces? Surely there are more level sets than that?
The kernel for general maps between sets is an equivalence relation: if $f\colon X\to Y$, then the kernel is the equivalence relation $\sim_f$ defined by $$ a\sim_f b\text{ if and only if }f(a)=f(b) $$ The Wikipedia page identifies this relation with the partition induced by it: $$ \bigl\{ \{w\in X:f(x)=f(w)\}:x\in X \bigr\} $$ where, for $x\in X$, $\{w\in X:f(x)=f(w)\}=\{w\in X:x\sim_f w\}$ is the equivalence class (or level set) of $x$.
When linear maps are concerned, there's a better description, because when $X$ and $Y$ are vector spaces and $f$ is linear, $$ f(a)=f(b)\text{ if and only if }f(a-b)=0 $$ so the kernel can be described just by the vector subspace $N=\{x\in X:f(x)=0\}$.
There's no real difference, except that in vector spaces (but also in groups or rings) the description of the kernel is handier.
The key fact is that $\sim_f$ is more than an equivalence relation: it is a congruence, that is, an equivalence relation that preserves the operations on the structure: if $a\sim_f b$ and $a'\sim_f b'$, then $$ a+a'\sim_f b+b' $$ and, when $\gamma$ is a scalar, also $\gamma a\sim_f \gamma b$.
The fact that a vector space (and likewise a group or a ring) has an operation which is associative, with neutral element and inverses, allows for the simpler description in terms of a single subset rather than with a partition.