Postgres 16

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select version();

version

PostgreSQL 16.0 (Debian 16.0-1.pgdg120+1) on x86_64-pc-linux-gnu, compiled by gcc (Debian 12.2.0-14) 12.2.0, 64-bit

SELECT 1

create table table_name (
    data jsonb null,
    column_name text null
); 

insert into table_name (data, column_name)
     values ('{"object_key_name": "hello_world"}', null)
           ,('{"object_key_name": "goodby_world"}', 10); 
           
select * 
  from table_name;

CREATE TABLE

INSERT 0 2

data
column_name

{"object_key_name": "hello_world"}
null

{"object_key_name": "goodby_world"}
10

SELECT 2

select * 
  from table_name 
 where column_name = null;

data
column_name

SELECT 0

 select * 
  from table_name 
 where column_name <> null;

data
column_name

SELECT 0

select *
  from table_name 
 where column_name is  null

data
column_name

{"object_key_name": "hello_world"}
null

SELECT 1

select *
  from table_name 
 where column_name is not null;

data
column_name

{"object_key_name": "goodby_world"}
10

SELECT 1

-- your test
update table_name
   set column_name = data ->> 'object_key_name'
 where column_name is null;

select * 
  from table_name;

UPDATE 1

data
column_name

{"object_key_name": "goodby_world"}
10

{"object_key_name": "hello_world"}
hello_world

SELECT 2

insert into table_name (data, column_name)
     values ('{"object_key_name": "nothing here"}', null); 

update table_name
   set column_name = data ->> 'object_key_name'
 where coalesce(column_name, 'it''null folks') = 'it''null folks';

select * 
  from table_name;

INSERT 0 1

UPDATE 1

data
column_name

{"object_key_name": "goodby_world"}
10

{"object_key_name": "hello_world"}
hello_world

{"object_key_name": "nothing here"}
nothing here

SELECT 3

data	column_name
{"object_key_name": "hello_world"}	null
{"object_key_name": "goodby_world"}	10

data	column_name
{"object_key_name": "goodby_world"}	10
{"object_key_name": "hello_world"}	hello_world