Is there a way to give a subquery an alias in Oracle 11g SQL?

Your query should be fine.

An alternative would be:

select abc.client_ref_id, abc.request, def.response
from   some_table abc,
       some_table def
where  abc.client_ref_id = def.client_ref_id
and    abc.message_type = 1
and    def.message_type = 2;

I wouldn't be surprised if Oracle rewrote the queries so that the plan would be the same anyway.

You can give a query a name or alias with CTE’s (Common Table Expressions) aka WITH clause aka by Oracle as Subquery Factoring:

WITH abc as (select client_ref_id, request from some_table where message_type = 1)
select * 
from abc
    inner join 
    (select client_ref_id, response  from some_table where message_type = 2) defg
       on abc.client_ref_id = def.client_ref_id;

I don't have an Oracle instance to test with, but what you posted should be valid ANSI-89 JOIN syntax. Here it is in ANSI-92:

SELECT *
  FROM (SELECT client_ref_id, request 
          FROM SOME_TABLE 
         WHERE message_type = 1) abc
  JOIN (SELECT client_ref_id, request 
          FROM SOME_TABLE 
         WHERE message_type = 1) defg ON defg.client_ref_id = abc.client_ref_id