The problem a solo faces delivering to several clients exists in an agency too, only amplified: several people can deliver under the same name, with different habits, which multiplies the chances of error (wrong link sent, wrong account cc'd).
One delivery procedure for the whole team
The real lever in an agency isn't individual vigilance, it's the procedure. A single standardised way to deliver (same tool, same naming structure, same checks) cuts human error far more than a reminder to be careful.
Access partitioned by client, not by memory
Never rely on the team's memory to know "who has access to what". Each client needs its own isolated delivery space, with its own links, so that a mistake on one account can never expose another.
On xFer, each delivery is a standalone page: a link created for one client technically can't expose another client's files, even when the whole team uses the same tool day to day.
A checkpoint before the final send
For sensitive deliveries (a big account, confidential content), add a second check before you send for good: another person confirms the right client, the right file, the right recipient. It takes two minutes and it prevents the incident that costs a client relationship.
In short
One procedure for the whole team, access partitioned by client, and a check before sending on sensitive accounts. In an agency, confusion is prevented by the system, not by individual vigilance.