If not defined via UI.

The answer from https://blender.stackexchange.com/a/24004/123064 used in other answer will work for properties defined by the UI, or with the "_RNA_UI" key set. used for adding extra details to the property like min, max description, etc_

It will not pick up on properties defined using

ob["foo"] = 3

Using ob.items() gives us a list of key, value pairs. The key being the name of the prop.

Test script,

import bpy

context = bpy.context
    
ob = context.object
# all props with name starting with "Shake_" on context ob
props = [(k, v) for k, v in ob.items() if k.startswith("Shake_")]

# all selected apart from ob
obs = context.selected_objects # selected
# obs = context.scene.objects[:] # all in scene
# obs = context.collection.objects # all in context collection...
obs.remove(ob)
# set from key value pairs
for o in obs:
    for k, v in props:
        o[k] = v
    

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If not defined via UI.

The answer from https://blender.stackexchange.com/a/24004/123064 used in other answer will work for properties defined by the UI, or with the "_RNA_UI" key set. It is used for adding extra details to the property like min, max description, etc

It will not pick up on properties defined using

ob["foo"] = 3

Using ob.items() gives us a list of key, value pairs. The key being the name of the prop.

Test script,

import bpy

context = bpy.context
    
ob = context.object
# all props with name starting with "Shake_" on context ob
props = [(k, v) for k, v in ob.items() if k.startswith("Shake_")]

# all selected apart from ob
obs = context.selected_objects # selected
# obs = context.scene.objects[:] # all in scene
# obs = context.collection.objects[:] # all in context collection...
if ob in obs:
    obs.remove(ob)
# set from key value pairs
for o in obs:
    for k, v in props:
        o[k] = v
    

Related.

How to edit a custom property in a python script?

How can I save a dict in a scene propertyGroup?

how to change the value of various custom properties at the same time?

Get all custom properties of an object

When should custom-properties be used instead of 'bpy.props'?

Update a custom data field created with an addon