Oracle 11g Release 2

with your_table (obj, obj_characteristic, characteristic_date_added) as (
            select 'Mark', 1, date '2018-01-15' from dual
  union all select 'Mark', 2, date '2018-02-15' from dual
  union all select 'Jimmy', 1, date '2018-01-31' from dual
  union all select 'Jimmy', 2, date '2018-02-11' from dual
  union all select 'Jimmy', 4, date '2018-03-15' from dual
  union all select 'Jimmy', 5, date '2018-04-15' from dual
  union all select 'Jimmy', 6, date '2018-04-15' from dual
  union all select 'Harry', 1, date '2018-01-08' from dual
  union all select 'Harry', 2, date '2018-01-11' from dual
  union all select 'Harry', 3, date '2018-02-15' from dual
  union all select 'Michael', 1, date '2018-06-15' from dual
  union all select 'Michael', 2, date '2018-07-15' from dual
  union all select 'Dwayne', 4, date '2018-01-15' from dual
  union all select 'Dwayne', 5, date '2018-01-15' from dual
  union all select 'Dwayne', 6, date '2018-01-15' from dual
)
select obj, obj_characteristic, characteristic_date_added
from (
  select obj, obj_characteristic, characteristic_date_added,
    count(distinct obj_characteristic) over (partition by obj) as c1,
    count(distinct case when obj_characteristic in (1,2) then obj_characteristic end)
      over (partition by obj) as c2
  from your_table
)
where c1 = c2;

OBJ	OBJ_CHARACTERISTIC	CHARACTERISTIC_DATE_ADDED
Mark	1	15-JAN-18
Mark	2	15-FEB-18
Michael	1	15-JUN-18
Michael	2	15-JUL-18

with your_table (obj, obj_characteristic, characteristic_date_added) as (
            select 'Mark', 1, date '2018-01-15' from dual
  union all select 'Mark', 2, date '2018-02-15' from dual
  union all select 'Jimmy', 1, date '2018-01-31' from dual
  union all select 'Jimmy', 2, date '2018-02-11' from dual
  union all select 'Jimmy', 4, date '2018-03-15' from dual
  union all select 'Jimmy', 5, date '2018-04-15' from dual
  union all select 'Jimmy', 6, date '2018-04-15' from dual
  union all select 'Harry', 1, date '2018-01-08' from dual
  union all select 'Harry', 2, date '2018-01-11' from dual
  union all select 'Harry', 3, date '2018-02-15' from dual
  union all select 'Michael', 1, date '2018-06-15' from dual
  union all select 'Michael', 2, date '2018-07-15' from dual
  union all select 'Dwayne', 4, date '2018-01-15' from dual
  union all select 'Dwayne', 5, date '2018-01-15' from dual
  union all select 'Dwayne', 6, date '2018-01-15' from dual
)
select distinct obj
from (
  select obj, obj_characteristic, characteristic_date_added,
    count(distinct obj_characteristic) over (partition by obj) as c1,
    count(distinct case when obj_characteristic in (1,2) then obj_characteristic end)
      over (partition by obj) as c2
  from your_table
)
where c1 = c2;

OBJ
Mark
Michael

with your_table (obj, obj_characteristic, characteristic_date_added) as (
            select 'Mark', 1, date '2018-01-15' from dual
  union all select 'Mark', 2, date '2018-02-15' from dual
  union all select 'Jimmy', 1, date '2018-01-31' from dual
  union all select 'Jimmy', 2, date '2018-02-11' from dual
  union all select 'Jimmy', 4, date '2018-03-15' from dual
  union all select 'Jimmy', 5, date '2018-04-15' from dual
  union all select 'Jimmy', 6, date '2018-04-15' from dual
  union all select 'Harry', 1, date '2018-01-08' from dual
  union all select 'Harry', 2, date '2018-01-11' from dual
  union all select 'Harry', 3, date '2018-02-15' from dual
  union all select 'Michael', 1, date '2018-06-15' from dual
  union all select 'Michael', 2, date '2018-07-15' from dual
  union all select 'Dwayne', 4, date '2018-01-15' from dual
  union all select 'Dwayne', 5, date '2018-01-15' from dual
  union all select 'Dwayne', 6, date '2018-01-15' from dual
)
select obj
from your_table
group by obj
having count(distinct obj_characteristic)
  = count(distinct case when obj_characteristic in (1,2) then obj_characteristic end);

OBJ
Mark
Michael