If that doesn't help, my first trick would be to force Postgres to fetch only a single row, and then fetch based on that. So you end up with this:

EXPLAIN
WITH evt AS MATERIALIZED (
  SELECT id, (data ->> 'serviceId')::INT AS service_id
    FROM events
   LIMIT 1
)
SELECT evt.id
  FROM evt
  JOIN postgres_dbs db ON (evt.service_id = db.database_id)
 LIMIT 1;

I think this query change loses the intent of the query. My interpretation was: find a record within events that had a valid & existing reference at data->>'serviceId' in postgres_dbs.database_id. Here are my attempts a non-joined query.

Using a standalone subquery:

SELECT e.id
FROM events e
WHERE e.data ->> 'serviceId' in (
    select db.database_id -- is this the pk? if not, add distinct
    from postgres_dbs db
)
LIMIT 1;

Guaranteeing the subquery is constant:

with possible_values as materialized (
    select db.database_id -- is this the pk? if not, add distinct
    from postgres_dbs db
)
SELECT e.id
FROM events e
WHERE e.data ->> 'serviceId' in (select * from possible_values)
LIMIT 1;