In the analytical geometry version of Euclidean Geometry we need a convenient way of thinking of parallelism and there is also the interest in constructing the real projective plane (where there are no parallel lines) from Euclidean Geometry. (The key idea is that we think of all the lines which have the same slope as having a special additional point on them at "infinity" (thereby creating new lines in the projective plane) and all these new points, one for each slope, lie on a new line, the line at "infinity.") In the analytical geometry framework of Euclidean Geometry lines are parallel when they have the same slope but there is the problem that a nonzero multiple of an equation looks like a different equation but consists of all of the same points. Furthermore, one would like parallelism to be an equivalence relation, which means it obeys being reflexive, symmetric and transitive.
So it is CONVENIENT to think of a line as being parallel to itself. Now all lines with the same slope can be thought of as parallel. (All the lines with undefined slope are also parallel to each other; they are the vertical lines.) Furthermore, we can think of the Euclidean plane's lines as being partitioned in classes by this parallelism relation.