Now you've got me wondering something: if a "stopwatch month" can't exist since everyone agrees that different months have different durations (and therefore you must select one like "the month of January" to know how long to run the stopwatch), isn't there an argument that a "stopwatch year" has the same need to select one since everyone agrees that different years have different day counts (unless we mean a solar year in seconds, not quantized to the nearest day, but that's probably a Bad Default)?

The collective human decision to make days-per-year vary (requiring leap rules to calculate days) seems similar to the collective human decision to make days-per-month vary (requiring month names to calculate days). So if we say a "stopwatch year" suffers the same fate as the "stopwatch month" then it's a slippery slope to saying the "stopwatch minute" is no different than a "stopwatch year" (requiring leap rules to calculate seconds) even if, for all practical purposes, it seems exempt.

I guess this is why we make "second" the SI unit, and none of our human-convenience rules mess with the duration of a second. A leap second changes the duration of a minute (and above), and a leap year changes the duration of a month (and above). Which, oddly enough, demonstrates an inconsistency: we ought to say "leap day" instead of "leap year" if the thing being added will follow the word "leap."