Can I set PostgreSQL "stringtype=unspecified" behavior as default?
Unfortunately the answer is no. 
stringtype is not server parameter. It's JDBC's own client parameter. 
Check the function here; 

org.postgresql.jdbc.PgConnection.getStringVarcharFlag()


and here

org.postgresql.jdbc.PgPreparedStatement.getStringType()


You will see that JDBC sets parameter type to VARCHAR or UNSPECIFIED type according to the stringtype property.