Hi! When I load an .obj file into houdini, it has multiple groups, one for each separate objects. But when I'm saving geometry(with file node) it saves it as one chunk.

Is there a way to save geometry with separate groups into an .obj file that has separate objects(named as groups) ?

Sure, just create vertical stacks for the objects and make sure the object in that stack belongs to a unique group.
Each object must be a true polygon object, i.e. not a primitive in the case of a sphere for example.

Using this technique gave me separate objects in Blender when I imported the OBJ file.
In Houdini you get a single object, but the group information is there.
I followed it up with a Delete node in non-selected mode for each given group to restore each object.

NOTE: Material colors do not transfer using this technique, use a better file format such as FBX or Alembic if you need to transfer more than geometry.