This is a limitation of the tools. The output line segments are not only in an arbitrary order, but they may also be multipart and non-contiguous.
The Identity tool might be used in a case where a line feature representing a road crosses multiple section/township polygon boundaries, and the task is to assign section/township IDs to the portions of the line passing through each polygon. The output line feature class contains line segments in a random order, not a progression of segments from the start of the original line feature to its end. The original line may turn and pass through a polygon more than once, in which case an output segment associated with that polygon may be mutipart with the parts not touching end-to-end.
To reorganize the line segments, they need to be rewritten in the desired order to a new feature class. For instance, in a script, the beginning vertex of all the line segments output from the Identity tool might be compared to see which one coincides with the original line feature beginning vertex (call that segment X). Write X to a new output feature class. Then examine the beginning vertex of all the remaining Identity output segments to see which one coincides with the ending vertex of X, write that to the new feature class, and so on.
Since the output of the Identity tool can be multipart, it is necessary to explode (run the 'Multipart to Singlepart' tool on) the Identity output data before running the workflow described above.