Postgres 12

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.

with dataset as (
  select 
    1       as id
  , 'alice' as name
  , 'a'     as key
  , 0       as metric
  union select
    2       as id
  , 'alice' as name
  , 'b'     as key
  , 1       as metric
  union select
    3       as id
  , 'alice' as name
  , 'b'     as key
  , 1       as metric
  union select 
    4       as id
  , 'alice' as name
  , 'a'     as key
  , 1       as metric
  union select
    5       as id
  , 'alice' as name
  , 'b'     as key
  , 0       as metric
  union select
    6       as id
  , 'alice' as name
  , 'b'     as key
  , 0       as metric
  union select
    7       as id
  , 'alice' as name
  , 'b'     as key
  , 0       as metric
  union select
    8       as id
  , 'bob'   as name
  , 'a'     as key
  , 1       as metric
  union select
    9       as id
  , 'bob'   as name
  , 'b'     as key
  , 1       as metric
  union select
    10      as id
  , 'bob'   as name
  , 'a'     as key
  , 0       as metric
)
select min(id) id, name, sum(metric) metric
from (
  select *, sum((key = 'a')::int) over (partition by name order by id) grp 
  from dataset
) t
group by name, grp
order by id

id	name	metric
1	alice	2
4	alice	1
8	bob	2
10	bob	0