Example 1 A line $l$ on the Cartesian plane is called a standard line if it is parallel or coincident with the $x$-axis, $y$-axis, or the lines $y = \pm x$. Given any 6 points on the Cartesian plane, connecting any two points forms a line. What is the maximum number of standard lines among these lines?
Let the set of 6 points given on a plane be $A$, and draw a line through any two points in $A$. Let there be $k$ regular lines, forming the set $B$. If a point $a$ in $A$ lies on a regular line $l$ in $B$, then pair $a$ with $l$, and denote this pair subset as $S$. On one hand, for any $a \in A$, there are at most 4 regular lines passing through $a$, and $|A|=6$, so
$|S| \leqslant 4|A|=24$.
On the other hand, for any regular line $l$ in $B$, $l$ has at least two points from $A$, and $|B|=k$, so
$|S| \geqslant 2|B|=2 k$.
Therefore, $2 k \leqslant 24, k \leqslant 12$.
If $k=12$, then $|S|=2 k=4 \times 6$, meaning that through each point in $A$ there are exactly 4 regular lines, and each regular line has exactly two points from $A$. Since $A$ is a finite set of points, there exists a rectangular region $D$ whose two pairs of opposite sides are parallel to the $x$-axis and $y$-axis, respectively, and such that all 6 points in $A$ are on the boundary or inside $D$, and each side of $D$ has at least one point from $A$. Thus, each side of $D$ is a regular line, and $D$ must have a vertex $a$ that is a point in $A$ (otherwise, each side of $D$ would have two different points from $A$, and the 4 sides would have 8 different points from $A$, which is a contradiction). Among the 4 regular lines passing through vertex $a$, there must be one that intersects $D$ at only one point, meaning that $l$ has exactly one point from $A$, which contradicts the fact that each regular line has two points from $A$. Therefore, $k \leqslant 11$.

On the other hand, as shown in Figure 4-7, the 6 points $a_{1}, a_{2}, \cdots, a_{6}$ have exactly 11 regular lines, each passing through two of them.
Therefore, the maximum number of regular lines connecting any two of the 6 points is 11.