Postgres SQL - How do I aggregated groups of multiple consecutive rows?

You can use a gaps-and-islands approach by marking the first record of each group when either there is no previous record for the tag or the v_start is greater than v_stop of the previous record:

select tag, v_start, v_stop, 
         coalesce(lag(v_stop) over w < v_start, true) as is_end_grp
    from data
  window w as (partition by tag order by v_start)

Use a windowed sum() of the boolean is_end_grp cast to int (1 if true, 0 if false) to number the groups:

  select tag, sum(is_end_grp::int) over (partition by tag 
                                             order by v_start) as grp_num,
         v_start, v_stop
    from mark_gaps

Aggregation over (tag, grp_num) will produce your desired result:

select tag, min(v_start) as v_start, max(v_stop) as v_stop
  from numbered_groups
 group by tag, grp_num
 order by tag, v_start

Working DB<>Fiddle

Using the numbered_groups logic from @Mike Organek answer. I just started from a different place

WITH data AS (
    SELECT *
    , case when lead(v_start) over(partition by tag order by v_start) = v_stop then 0 else 1 end stopcheck
    , case when lag(v_stop) over(partition by tag order by v_stop) = v_start then 0 else 1 end startcheck
    FROM (VALUES 
        ('a' , 2 , 3),
        ('a', 3, 5),
        ('a', 5, 7),
        ('a', 8, 10),
        ('a', 10, 12),
        ('a', 12, 14),
        ('b', 7, 8),
        ('b', 8, 10),
        ('b', 12, 15),
        ('c', 10, 11)
    ) AS T(tag, v_start, v_stop)
    ORDER BY tag, v_start, v_stop
)
,cnt as (
  select *
  , sum(startcheck) over (partition by tag order by v_start) grpn 
  from data)
select c1.tag, c1.v_start, c2.v_stop
from cnt c1 
inner join cnt c2 
  on c1.tag = c2.tag and c1.grpn = c2.grpn
  where c1.startcheck = 1 and c2.stopcheck = 1

This logic is all based on the assumption that your data always starts where the last row left off, there is no overlap etc.

Create a startcheck and stopcheck by comparing the prior row and next row relatively. From here use another window function sum() over to order the start records (so we don't match start of second batch to stop of first batch)

Join the table to itself matching like tag and groups. Filtering start and stop records