The calculator should then show "approximately N months" -- showing "N months and M days" is nonsensical if "M" has no useful precision
In the context of the calculator, you already know the start and end times, and it shows the duration in terms of days (which is unambiguous given that we know the interval when the duration takes place). I think that it's clear that the duration in terms of months, weeks and hours is there just to put it in more easily graspable terms, at the expense of precision.
The truth in both cases is of course that while the meaning is contextual and imprecise, it's still useful because people generally don't use precise measurements in their day-to-day activities. When someone tells me that something will happen in a month, I take it to mean something in the ballpark of 28-31 days, and that information can be useful to me.
Because of leap seconds, the only really unexceptional unit of time is seconds, but someone telling me that something will happen in 2592000 seconds isn't particularly useful to me.