I have a mesh imported from an STL file. Although it is 1 mesh, it consists of 2 objects, for example in the image below, one mesh of two spheres.

In this image, although it is 2 spheres, it is actually a mesh object.
I want after using Mesh Slicer, the result I will get 2 mesh objects as shown below.

Hi. There should not be any issues to slice such mesh. One thing you could have a problem with is if your object have two colliders and the "Slice" plane does not touch it:
You have two options here:
1) make one big collider:
2) or use manual slicing. It do not require any collider to exists. An example could be found in sample scene here: testSceneManual.unity