Postgresql update a partitioned table, where update affects multiple partitions

I was wondering should I separate table like below, or not?

Well, one immediately obvious downside of the table design you posted is that there are no constraints on user_information or user_role ensuring that you don't have more than one such row in each of these tables for each user. And even if you remember to add that, you will still have the problem that it will be difficult or impossible to enforce the rule "every user must have a role (or other critical attribute)".

Where it typically makes sense to break out metadata like this is when you have either many-to-one relationship of this metadata (e.g. a single user may have multiple roles), or you have a lot (i.e. many columns) of optional metadata that is unwieldy to cram into a single table.

For now I only can imagine I might want to know each column last modified time?

So if I want to know each column last modified time, is there any original build method in PostgreSQL? or I have to add for each column like email_last_modified_date, username_last_modified_date ...

You'll have to create individual columns (e.g. "email_last_modified_date") and have a trigger function enforce that the column is set correctly to achieve this. A more flexible and useful approach is usually constructing an audit table, with a trigger on the "user" table saving the changed columns for every UPDATE, or the contents of the whole row for every DELETE. This is where json, jsonb, or hstore types come in handy, if you'd like a single audit table capable of dealing with several tables, or capable of dealing with many columns from a single user table without needing to know the structure of the user table.

It seems there is a UNIQUE INDEX on table A(user_id) and you're trying to assign an existing value to more than one row.

Have a look at the next example:

create table a (id int, user_id int, f_id int);
create table b (id int, f_id int);
create unique index A_user_id_key on a(user_id);

insert into a 
values (1, 1, 11),
       (2, 3, 22),
       (3, 2, 33),
       (4, 5, 44 );

insert into b
values (1, 11),
       (3, 22),
       (3, 33), --<<<< there is another row with user_id = 3
       (4, 44);

When I try to update using your current query:

update a
set    user_id = b.id
from   b
where  a.f_id = b.f_id;

It returns same error message:

ERROR: duplicate key value violates unique constraint "a_user_id_key"
DETAIL: Key (user_id)=(3) already exists.

You could solve by avoiding existing values:

update a
set    user_id = b.id
from   b
where  a.f_id = b.f_id
and    not exists(select 1 from a where a.user_id = b.id);

This is the result:

select * from a;

id | user_id | f_id
-: | ------: | ---:
 1 |       1 |   11
 2 |       3 |   22
 3 |       2 |   33
 4 |       4 |   44

db<>fiddle here

But I'll suggest to check wich are the duplicate rows using next query:

select a.*
from   a
join   b
on     a.f_id = b.f_id
and    exists(select 1 from a where user_id = b.id and f_id <> b.f_id);

id | user_id | f_id
-: | ------: | ---:
 3 |       2 |   33

db<>fiddle here