Understand your suspicion!
It is increasingly common for employers to try and move away from relying on interviews alone as a means of testing candidates as it's been statistically shown that interviews are not a good way of predicting/selecting people who will be the best at doing the job, especially in roles that are more technical, skill based in nature.
So lots of companies are looking at ways of using more testing and replication of the demands of the actual role as part of the selection process. Most commonly this is through assessment centres where you have to do in-tray exercises, skills tests, role plays etc, but even this has been shown by research to not replicate how people will actually be in the workplace as it's such an artificial environment. So it seems we need to move towards observing people in a real-world situation and with the actual people/in the environment they'll be in to get an accurate idea of their future performance, which might be what they're trying to do through the 'do the role for a day' they've mentioned? I have to say I've not really heard of a large employer doing this systematically and successfully as part of their recruitment processes yet but it could turn out to be absolutely standard practice in a few years, or it could be the latest idea to be thrown on the HR scrapheap, who knows! This could be a really pioneering employer utilising super-innovative recruitment practice!
However, it's also not unheard of for employers to try and score some free labour by making candidates do unpaid 'trial' shifts but I've only really heard of this in low skill type roles like retail and catering rather than senior leadership positions. As you say I would have thought there's not much tangible you could achieve in a day and if there's commercial sensitivity they possibly stand to lose more than they'd gain anyway?
I'm always more inclined to attribute things like this to over-enthusiasm for a new idea/ignorance of usual processes than conspiracy, so if I was you I think I would be inclined to wait and see whether this actually materialises and if so, respond neutrally asking for more details of what would actually be expected of you during this day, how they will be assessing you and what the practical arrangements would be, and take it from there depending on how much you want the job. It does seem a bit much to expect a full day from you when you've already been in for two interviews (it will presumably mean taking a day's leave? and they aren't expecting to pay you?) but if it's a dream job then I guess I'd do it...