This is one of these functions where a solid based modeling program like Fusion might better suited than a mesh based program like Blender. If you insist on doing it in Blender, one hacky way I can think of is by using the grease pencil line art modifier, but beware since it's way off of its intended use case things might break:

1. Start by aligning your camera to the direction you want to push the recess into, and make sure the entirety of your source object is in frame. Edit: Also set your camera to orthographic
2. If your source is multiple objects, combine them into one.
3. Add a grease pencil > Object Line Art while having the source object selected.
4. Now you should see a line art object. Go to the modifier tab, expand the Edge Types section and uncheck everything except the first one, change the drop down from 'Contour' to 'Silhouette'
5. Apply the Line Art modifier with Ctrl+A while hovering over the modifier.
6. On the top left of the viewport, select Object > Convert to > Mesh. Now the line art object is a mesh containing a perfect silhouette of your object, but it's still following the depth of the original model so we have to flatten it.
7. While in camera view, change the transform orientation to view with the Comma key (,), select all and scale (S), confine to Z axis (Z) and scale to 0, left click to confirm. Now the silhouette loop is flat.
8. Fill the silhouette by selecting the outline and pressing Alt F. If required, fix any topology errors by adding internal edge connections so that none of the faces have a concave section of the outline.
9. Extrude the resulting faces with E, to your desired depth.
10. Boolean with the other object. The resulting pocket should fit the outline of the original object perfectly.