Gift Box uses draft orders to send the gifts to checkout, and Shopify multi-currency was built to support only the online store sales channel, but does not support draft orders. This means that, if you are using Multi-Currency in your store as well as our Gift Box app, your customers will see the default currency from your store at checkout once they become eligible for a gift. So, at the moment, both are incompatible.
We know Shopify is working on getting the Multi-currency compatible with draft orders, but we do not have an estimated time for the release. If this happens in the future, we certainly will notify our users.
You can see more information here within the Shopify community for a better understanding of the situation.