@Kit29,
No, this is not normal, and it sounds like another example of a CS rep making/mixing up information. New hosts may experience a delay with their first long-term stay payout, but doesn't apply to experienced hosts. Payouts for multiple stays that occurred within a short period may receive one combined payout, or a host may request that payout are made after a minimum amount has been made.
We regularly host long-term stays, and payment is sent 24-48 days after check in, and then every 30 days. Airbnb attempts to collect payment from guests approximately 10 days before their next stay period begins. If there is a problem with payment, ABB notifies the hosts and informs them that if the payment hasn't been made 24 hours prior to the end of the paid reservation period the host can cancel the remainder of the reservation without penalty.