A simple solution
The simple solution assumes objects don't move and you never have objects with same location (or you're fine with them synchronizing the random values in such case):


If you want to be able to move those objects without rerandomization (for animation purposes), you could use a Simulation Zone:


XY's solution

random_type = 0
"""
random_type = 0 - same behavior as "Random per Object" in shaders, value depends on object name
random_type = 1 - as above but multiple modifiers on the same object
random_type = 2 - the script randomizes different numbers for the same object each time it is run
"""
from bpy import context as C, data as D
from random import seed, randint
randomizer = D.node_groups['Randomizer']
input_id = next(i.identifier for i in randomizer.inputs if i.name == 'Seed')
for ob in C.selected_objects:
if random_type == 1:
seed(ob.name)
mods = (m for m in ob.modifiers if m.node_group == randomizer)
for mod in mods:
if random_type == 0:
seed(ob.name)
random = randint(-231, 231-1)
mod[input_id] = random
ob.data.update()

- Random Value node is there, because Noise Texture's $W$ dimension is limited to $-1000...+1000$ range [it isn't! Only the interface limits the input in that range; Learned it thanks to folks on Discord]; the code could be modified to generate a random float in that range, but this way randomizing seeds is decoupled from the node trees using those seeds.
- The script has to be run each time you add a new object, and - if you want the seeds to be synchronized with object names - each time you rename an object.
- You need to select all relevant objects before running the script but that could be trivially changed if needed…
- Notice how I usually don't screenshot a node tree name, but in this case I did because that's how the script discriminates the correct modifiers: there could be Geonodes modifiers with a seed input that is not supposed to be randomized.
https://blender.stackexchange.com/questions/298986/geometry-nodes-make-all-groups-single-user-at-once/298988#298988 – X Y Aug 16 '23 at 06:34
https://blender.stackexchange.com/questions/298795/how-do-i-place-a-random-object-from-a-specific-collection/298800#298800 – X Y Aug 16 '23 at 06:38