How do I programmatically determine if an object is a 'linked duplicate' of another?

Object.data

Linked duplicates will have the same linked data, if there is linked data. For an empty ob.data is None

import bpy
from collections import defaultdict

scene = bpy.context.scene

data_links = defaultdict(list)
for ob in scene.objects:
    data_links[ob.data].append(ob)
    
for k, v in data_links.items():
    print(k, v)

Sample output, notice there are 4 objects using the mesh "Cube.001"

<bpy_struct, Mesh("Cube.001")> [bpy.data.objects['Cube'], bpy.data.objects['Cube.001'], bpy.data.objects['Cube.002'], bpy.data.objects['Cube.003']]
<bpy_struct, PointLight("Lamp")> [bpy.data.objects['Lamp']]
<bpy_struct, Camera("Camera")> [bpy.data.objects['Camera']]
None [bpy.data.objects['Empty'], bpy.data.objects['Field']]