Name for genderless/male-and-female-all-in-one connectors for chaining PCBs?
Hermaphroditic connectors. Most I see are vertically hermaphroditic though and you seem to want horizontally hermaphroditic. Note you can turn the board upside down and plug it in the wrong way, though you can do that in your original image too.
With regards to reducing BOM count, most hermaphroditic connectors seem to be vertically hermaproditic (like interlocking L-shapes that sit vertically on the board). In this case, unless it is truly an edge-mounted version, would need two different versions (right side up and upside down) to mount on the same side of the PCB in order to not produce a stair-case of PCBs when chaining boards together. So you would still need two parts on your BOM.
However, if you can find horizontally hermaphroditic connectors (like interlocking L-shapes that lay flat in the plane of the board) that basically look like single-part versions of your male/female connectors then the stair case is not an issue with just a single part number.