The logic is that it is named {object name}_{mesh name}, unless the object name and mesh name are the same, in which case their common name is used. So you can avoid this by renaming the mesh on the "Cube_HP" object to "Cube_HP" too.

The object name containing an underscore doesn't seem to matter.

Its not about the object name. My mesh is named "Cube_HP" when exporting it as a .obj. But upon importing that .obj it to any software or blender its name becomes, "Cube_HP_Cube" , which I dont want. It never happens to fbx. Thats why its a bug.
Even if u have a single mesh in ur blend file with name having a underscore somewhere it will always happen something like this to ur mesh name.

Your mesh is not named "Cube_HP", your object is. There is both an object and a mesh in Blender. They have different names. Yours are named like this, yes?

But there is not both an object and a mesh in .obj, so the exporter smushes the names together.

If you name both the object and the mesh "Cube_HP", then export, when you import the .obj, the imported object and mesh will both be named "Cube_HP".

Ohhhhh, wow, didn't know that name would have been used. My bad, Thank you for clarification.