create table clone_test( user_id integer, product_id integer);

insert into clone_test values (1,1), (1,2), (1,3), (2,2), (2,3);

insert into clone_test (select 3, product_id from clone_test where user_id =1);

select * from clone_test ;
 user_id | product_id 
---------+------------
       1 |          1
       1 |          2
       1 |          3
       2 |          2
       2 |          3
       3 |          1
       3 |          2
       3 |          3

copy all rows with user_id = 1 but change it to user_id = 3:

As already demonstrated by @Adrian Klaver, that’s a simple query with a constant in place of the user_id:
_{demo at db<>fiddle}

insert into your_table 
select 3 as user_id, product_id 
from your_table
where user_id=1;

Is there any way to do it without manually listing all other columns?

There is, but

• modern IDEs can typically be configured to identify your db schema, so they can add those columns on their own, without you having to type all of them out
• dynamic SQL can be used to query your information_schema to add those columns into your query, but that’s pretty ugly
• both dynamic SQL and the jsonb trick are quite costly and ugly, so it’s better to make sure benefits of this justify that price

You can use auto-mapping Postgres does when converting records to and from jsonb:

insert into clone_test 
select (jsonb_populate_record( null::clone_test
                              ,jsonb_set(to_jsonb(clone_test) 
                                         ,'{user_id}'
                                         ,to_jsonb(3))
                             )
       ).*
from clone_test
where user_id=1;

1. to_jsonb() grabs entire rows from the table and maps them to jsonb.
2. In each, jsonb_set() changes user_id to a 3.
3. jsonb_populate_record maps the updated jsonb back to a clone_test record.
4. ().* unpacks the record, separating the columns.

Doing all that just to avoid having to tab through auto-completion, type out or copy the column list, is an overkill.