In classical geometry why is a line considered to be parallel to itself?

The idea is that you want "parallel" to define equivalence classes (called "pencils", cf. Coxeter, Projective Geometry, and Artin, Geometric Algebra), which require the defining relationship to be an equivalence relationship: reflexive, symmetric, and transitive. Those classes then have some nifty uses, like defining projective space by adding a point at infinity for each pencil (which is the considered to be on each of those lines) and a line at infinity for each class of parallel planes (this line containing all the points at infinity corresponding to the pencils of lines in that class of planes).

Also, you were already going to have to rethink the definition of "parallel" as having no points in common, if you are going to do solid geometry. Parallel lines also need to be coplanar...i.e., there need to be two other lines that intersect each other and that each intersect the parallel lines (five distinct points of intersection). Lines that are not coplanar are called "skew" not "parallel".

If two lines are both parallel to a third line, they should be parallel to each other, even if they are the same line.

This is also quite natural when starting from the definition in a metric space:

Two lines $a$ and $b$ are parallel if for every point $p\in a$, the distance from $p$ to $b$ is the same.

Then, $a$ is parallel to itself because for every point $p\in a$, the distance to $a$ is zero.