If I have this correct, prize-linked savings accounts still generate some interest, but not as much as other savings accounts. In exchange for getting less interest, one is entered into monthly drawings for prizes. As such, I’m not entirely convinced this passes the threshold for gambling. Someone with a prize-linked account is not at risk of losing the money placed into the account, is not at risk of losing the money generated by the low interest rate, and is not at risk of losing anything else already owned. The element of chance here is found only in the fact that you may never get anything extra out of the account, whereas gambling comes with a risk of loss.
One might think that the lower interest rate is a loss. If it is, though, it is not a loss from risk. It is a known factor going into the arrangement, which is more properly called a “cost.” My checking account has a lower interest rate than my savings account, but it is not gambling for me to put money into it—even if I transfer money into it early because there is a chance one of my scheduled payments might drop before expected. Nor is it gambling for me to put money into my checking and savings accounts rather than accounts with a higher yield. Thus the argument against that this is gambling seems to be surmountable.
I don’t know that I want to call these sorts of costs “losses,” however, as I’m not really paying anything out just because I have an account with a lower interest rate. Moreover, a person whose will actually be choosing between “save nothing and get no interest” and “save money and get some interest” is not really thinking of accounts with a higher yield as a live option. Relative to the situation, then, there isn’t really any loss at all—just less of a gain than might be possible. This might also go to answering the argument against raised by @roundsquare.
This leaves the worry that the element of chance found in the prize lottery could be used to mask cheating. It’s a genuine worry, though one might wonder what its effects are supposed to be. Depending on how many of these accounts there are, the average person might never win a prize over the course of an entire life even under perfectly fair conditions. If the system is manipulated such that a person will never win a prize—perhaps because the actual winners are chosen by collusion among bank managers—the practical difference is nil. This does not make it acceptable, of course, but it does make the possibility of cheating less fearsome.
We could require some sort of review of the winners, though, or the process by which winners are chosen. That might reveal whether or not the prizes are being awarded to a particular demographic at a suspicious rate. I’d need to hear more about what kind of manipulation the system could be subject to before declaring this issue fully resolved, but it seems less worrisome so long as everyone is still keeping their money and interest. The prizes are an incentive, after all, not a guaranteed benefit.