Comparing multiple rows and column in oracle

You could write 2 queries, {Q1} returning the values you need for "Output 1", and {2} giving you the values for "Output 2". Once you see that these queries produce the correct result, JOIN them together. Example see DBfiddle.

Query 1 ("stage level")

select 
  projectno, stages, startdate, completiondate 
, case
    when startdate = completiondate then 'on time'
    else 'delayed'
  end output_1
from projects;

Query 2 ("project level")

-- look at the last stage (only).  CASE may need tweaking
select 
  projectno
, case 
    when max( startdate ) = max( completiondate ) then 'on time'
    else 'delayed'
  end output_2  
from projects
group by projectno
;

JOIN

select Q1.*, Q2.output_2
from (
  select 
    projectno, stages, startdate, completiondate 
  , case
      when startdate = completiondate then 'on time'
      else 'delayed'
    end output_1
  from projects
) Q1 join (
  select 
    projectno
  , case 
      when max( startdate ) = max( completiondate ) then 'on time'
      else 'delayed'
    end output_2  
  from projects
  group by projectno
) Q2 on Q1.projectno = Q2.projectno 
order by Q1.projectno, Q1.startdate
;

-- result
PROJECTNO   STAGES  STARTDATE   COMPLETIONDATE  OUTPUT_1  OUTPUT_2
PROJ_001    1      12-MAR-21    12-MAR-21       on time   on time
PROJ_001    2      14-MAR-21    14-MAR-21       on time   on time
PROJ_001    3      15-MAR-21    15-MAR-21       on time   on time
PROJ_001    4      18-MAR-21    18-MAR-21       on time   on time
PROJ_002    1      16-MAR-21    18-MAR-21       delayed   delayed
PROJ_002    2      17-MAR-21    19-MAR-21       delayed   delayed
PROJ_002    3      19-MAR-21    19-MAR-21       on time   delayed
PROJ_002    4      21-MAR-21    23-MAR-21       delayed   delayed

ADDENDUM

( Taking @Thorsten Kettner's suggestion on board: ) You could also use max() in form of an analytic function eg

-- remove the comments -> see the output of max(...) over (...)
select 
  projectno, stages, startdate, completiondate 
, case
    when startdate = completiondate then 'on time'
    else 'delayed'
  end output_1
, case 
    when 
      max( startdate ) over ( partition by projectno  )
    = max( completiondate )  over ( partition by projectno )
    then 'on time'
    else 'delayed'
  end output_2
-- , max( startdate ) over ( partition by projectno  ) maxstart_
-- , max( completiondate )  over ( partition by projectno ) maxcompletion_
from projects;

DBfiddle