How to use synonym of a DBlink in Oracle?

I don't see the point in creating a synonym for the dblink itself. Ideally you create the synonym for the remote table using the dblink.

CREATE DATABASE LINK my_db_link CONNECT TO user IDENTIFIED BY passwd USING 'alias';
CREATE SYNONYM my_table FOR remote_table@my_db_link;

Now, you could query the remote table using the synonym:

SELECT * FROM my_table;

Unfortunately creation of synonyms for dblinks is not supported. If you read the documentation on synonyms, you will find that the permitted objects for synonyms are only:

Use the CREATE SYNONYM statement to create a synonym, which is an alternative name for a table, view, sequence, procedure, stored function, package, materialized view, Java class schema object, user-defined object type, or another synonym.

The reason why your second query fails is that the synomym you have created is not functioning correctly. It is not being validated properly at creation time, and you can create any sort of incorrect synonyms like that. To verify, just test the following statement:

create synonym dblink3 for no_object_with_this_name;

You will still get a response like this:

*Synonym DBLINK3 created.*

But of course nothing will work via this synonym.