Commerce > Catalogue Items > Catalogue Item Edit > Gift > Gift Message
The (Gift) panel on a Catalogue Item has several options, one of those is Gift Message, which is intended for the Gift Recipient of a purchase.
When a purchase is made as a gift, the Gift Recipient will receive the corresponding notification on the day the gift (i.e. a subscription) starts, not on the day of purchase.
Note: the confirmation receipt is sent out on the date of purchase.
You can also create a conversion event that sends a message on the purchase date.