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Essentially I want to union multiple (20) congruent line shapefiles, where overlapping line segments in the input shapefiles are combined into a single resulting output shapefile. But I can't union lines. So I'm looking for an alternative. The desired result, like union, would break all the overlapping lines into separate features and preserve the input attributes.

Merge doesn't work for me because the inputs are preserved as separate (overlapping) features in the resulting shapefile. I tried dissolving this result, as well as Split Line at Vertices without getting the desired result.

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The topic that I think you are looking for is Linear Referencing and, to perform line-on-line overlay, the tool to use is Overlay Route Events.

I recommend reviewing About overlaying event data:

Overlaying events is another way to create new event data. This process combines two input event tables to create a single output event table. The new table can be used to analyze event data in ways not possible using traditional spatial analysis techniques.

The new event table can contain either the intersection or the union of the input events. The union of the input events splits all linear events at their intersections and writes them to the new event table. The intersection of the input event tables writes only overlapping events to the output event table.

Line-on-line, line-on-point (same as point-on-line), and even point-on-point event overlays can be performed.

Once you have the unioned linear events you should be able to convert them to features.

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For someone willing to consider any alternative (not just ArcPy), the Union geoprocessing tool in QGIS provides precisely the desired functionality. Within QGIS, go to Vector --> Geoprocessing Tools --> Union and select the two layers.

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