This is a gaps and islands problem. Let’s start by creating a table of test values and populating it:

CREATE TABLE test_values AS (
SELECT id, started_at::TIMESTAMP AS started_at
      FROM (VALUES (1, '2024-07-01 00:00:00'),
                   (2, '2024-07-01 00:10:00'),
                   (3, '2024-07-01 00:15:00'),
                   (4, '2024-07-01 00:19:00'),
                   (5, '2024-07-01 00:25:00')) v (id, started_at));

The following query demonstrates an approach to finding the starting times of each cluster:

WITH
  params AS (SELECT INTERVAL '5' MINUTE AS spread),
  ng AS (
    SELECT tv.id,
           tv.started_at,
           tv.started_at - LAG(tv.started_at) OVER (ORDER BY tv.started_at) > params.spread IS NOT FALSE AS new_group
      FROM params
      CROSS JOIN test_values tv)
SELECT ng.*
  FROM ng
  WHERE ng.new_group
  ORDER BY ng.started_at;

The query works by identifying each started_at that neither overlaps nor is contiguous with the most recent earlier time plus spread. The comparison is checked with IS NOT FALSE so that the first time will be identified as starting a group:

If > is changed to >=, then clusters will consist of times that are within spread of adjacent times; i.e, adjacent times that are exactly spread apart will be in different clusters: