In classical geometry why is a line considered to be parallel to itself? A definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is probably for convenience, but in my mind since two distinct lines are parallel if they have no points in common and a line has infinitely many points in common with itself. Perhaps the idea is to ease the definition that two (non-parallel) lines intersect at one and only one point? 
Q: What's the purpose/what inconvenience would be caused if we didn't have that definition?

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".
