Postgres 12

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.

with t (id, client_id, message, user_id, incoming) as (values
(1  , 1         , 'Hi, anybody there?'            , 2       , True),
(2  , 1         , 'I need help'                   , 2       , True),
(3  , 1         , 'Yes, I am here to help you.'   , 2       , False),
(4  , 2         , 'Did you solve it yet?'         , 5       , True),
(5  , 3         , 'Is my issue resolved?'         , 5       , True),
(6  , 2         , 'yes, it is solved'             , 5       , False),
(7  , 5         , 'Are you happy with us?'        , 3       , False),
(8  , 5         , 'yes, very much'                , 3       , True),
(9  , 1         , 'Hi, anybody there again?'      , 2       , True),
(10 , 1         , 'I need help'                   , 2       , True),
(11 , 1         , 'Yes, I am here to help you.'   , 2       , False),
(12 , 1         , 'Again.'                        , 2       , False)
), ch as (
  select t.*
       , case coalesce(incoming !=
             lag(incoming) over (
               partition by client_id, user_id 
               order by id
             )
             , true) 
           when true then 1
           else 0
         end as incoming_changed
  from t
), groups as (
  select ch.*
       , sum(incoming_changed) over (
           partition by client_id, user_id
           order by id
         ) as grp
  from ch
), grouped as (
  select client_id
       , string_agg(message, ' ') as message
       , user_id
       , incoming
       , min(id) as min_id
  from groups
  group by client_id, user_id, grp, incoming
  order by client_id, user_id, min_id
), paired as (
  select grouped.*
       , lead(min_id) over (
           partition by client_id, user_id
           order by min_id
         ) as response_id
  from grouped
)
select pc.client_id
     , pc.message as client_message
     , pu.user_id
     , pu.message as user_message
from paired pc
join paired pu on pc.response_id = pu.min_id
where pc.incoming

client_id	client_message	user_id	user_message
1	Hi, anybody there? I need help	2	Yes, I am here to help you.
1	Hi, anybody there again? I need help	2	Yes, I am here to help you. Again.
2	Did you solve it yet?	5	yes, it is solved