They're definitely making money: You pay them the full amount of the gift card upfront, in exchange for coffee later. That's an interest-free loan to Starbucks, and these have a monetary value these days!
> They "make" money when you leave a balance on the card.
In many US states, the money interestingly goes to the state in the end when unused, under a common law doctrine that doesn't exist in many other countries:
In that case, there is no breakage income for the gift card issuer, but the interest free loan, together with people's tendency to spend higher total amounts at the same merchant when using gift cards, still makes them an amazing deal to the issuer.
I suspect that there's also a non-negligible benefit being exploited in the form of differences in subjective value between gifter and giftee: In a nutshell, the gifter spends more money than they normally would at a store they frequent, or viewed from the giftee's perspective, they spend "money" at a company they normally wouldn't.
"Directly"? Isn't the point that they get to invest the money their customers add to their cards, for whatever time Starbucks hold it? That some customers also fail to redeem the balance is for them a bonus, but not what the "Starbucks is a bank" comment addresses.
Starbucks is another good example of one that does (with gift cards instead of points); Amazon might be another.