In the Order search window, the Patron name always matches the Recipient name and does not change with Gift memberships or Event registrations.
The “recipient” on the order is the person who it was “shipped” to, not the recipient of the gift. In Daily Sales, the recipient and patron will always be the same, since the orders are done in person. However, in Advance Sales, on the Add Payment section, you can specify a different recipient for the order than the patron, if needed. Then in Order Search, the patron and recipient will be different.