How do we define the distance between two lines or planes that intersect? How do we define the distance between two lines or planes that intersect? Since it cannot be defined injectively for every point, is it defined at all in such a case? 
 A: As noted in comments and other answers, the distance between them in the usual sense is $0$. Another way to measure "how far apart" they are is to use the angle between them. That would be $0$ if they were identical and $\pi/2$ if they were perpendicular. Perhaps that  would be useful in your application.
A: In this situation it is better to use the distance on the Grassmann manifold. First, given a set $E \subset \mathbb R^p$ and $v \in \mathbb R^p$, let
$$
d(v,E)=\inf \big\{\lVert v-w \rVert: w \in E \big\}.
$$
Then, given subspaces $E,F \subset \mathbb R^p$, we define
$$
d(E,F)=\max \bigg\{ \max_{v \in E, \lVert v \rVert=1} d(v,F), \max_{w \in F, \lVert w \rVert=1} d(w,E) \bigg\}.
$$
PS: We cannot use angles without anything else simply because they do not define a distance!
A: The distance between two sets that intersect is always $0$.
A: In a metric space, given a distance function $d$ on a space $X$, it is customary to define the distance between two nonempty subsets $A,B$ of $X$ as $d(A,B)=\inf(\{d(a,b)\mid (a,b)\in A\times B\})$.
In the special case of ordinary Euclidean distance, and two subsets that share a point $x$, the minimum would be attained by $d(x,x)=0$.
But using this definition, two sets that are disjoint can also have $d(A,B)=0$. For example, if $A=\{(a,0)\mid 0<a<1\}$ and $B=\{(1,0)\}$.
