Changing spatial extent of shapefile using ArcGIS Desktop?

If you are using ArcGIS Desktop 10.4 or later, and a license level of Standard or Advanced, then there is a tool available that enables you to Recalculate Feature Class Extent:

This tool recalculates the XY, Z, and M extent property of a feature class based on the features within the feature class.

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Recalculate Feature Class Extent updates geodatabase feature classes or shapefiles (point, multipoint, line, or polygon).

However, as you have discovered, it uses the extent of features within the feature class, rather than the extent of another feature class, to do that recalculation.

As others have commented, it is hard to envisage a use case where you would want the extent stored with a shapefile to be other than the features that it contains. If you still think that you do then I would recommend describing that use case in more detail within your question.

You can change the extent manually by right clicking on the layer in the catalog view --> properties --> "Feature Extent" Tab. (ArcGIS Desktop 10.6.1).

I had this same issue when calculating euclidean distance from a road, where the buffer would end abruptly at the edge of the road layer's extent (which was smaller than my DEM extent)

I think one of the shapefile is in UTM so you have to change its XY coordinate system from the catalog to UTM Zone... and then project it to the coordinate of the first shapefile.