Custom Blender 3.1+ FBX Exporter: Individually export all the selected objects and centers them on X and Y.

custom_exporter.py

# Individually export all the selected objects and centers them on X and Y.
#
# You have to provide the export folder path, if not provided, files
# will be exported under the same blend file directory under "/custom_export"
#
# Based on: https://blender.stackexchange.com/a/5383

import bpy
import os

# Absolute path to the folder where to export all the files.
# Windows e.g: "C:\\Users\\user\\exported_meshes"
export_folder_path = "C:\\Users\\my_user\\output_folder" # <-- Change this path!
# If the folder is not valid, export where the blend file is located.
if not os.path.isdir(export_folder_path):
    export_folder_path = bpy.path.abspath('//custom_export')
    if not os.path.isdir(export_folder_path):
        os.mkdir(export_folder_path)

# Get all the selected objects that are meshes.
selected_objects = [obj for obj in bpy.context.selected_objects if obj.type == 'MESH']
# Unselect everything since we will select one by one to individual export.
bpy.ops.object.select_all(action='DESELECT')

for object in selected_objects:
    # Store the original location.
    original_location = object.location.copy()
    # Move to x=0 and y=0 so the mesh is conviniently exported with the pivot in the middle.
    object.location = (0.0, 0.0, original_location.z)
    # Generate the file path where it will be exported (export_folder_path + object.name + .fbx)
    file_path = os.path.join(export_folder_path, object.name + ".fbx")
    # Select the current object to be exported.
    object.select_set(True)
    # Export as FBX.
    bpy.ops.export_scene.fbx(
        filepath=file_path,
        use_selection=True,
        # Add more export options if needed. Those can be found here:
        # https://docs.blender.org/api/dev/bpy.ops.export_scene.html
    )
    # Move the object back to its original location.
    object.location = original_location
    object.select_set(False)

# Select the original objects again so the scene is the same initial sate.
for object in selected_objects:
    object.select_set(True)