Stationary exercise bikes *can* measure power accurately, and some *do*. See the Cycleops 400 Pro, for example.
So perhaps your question is, rather, why some cannot accurately measure watts? I don't get why it matters, and the main answer was given early upthread: they have no tool to measure force, which as @
jsk noted, is commonly a set of strain guages for those that do. Wattbike use what they call a 'load cell', which I think is a collection of strain guages, and which they uniquely mount on the chain. Others mount on cranks and hubs.
There are, of course, other ways to measure watts, but I can only suppose they're difficult to imlement on a stationary bike, otherwise they'd exist.
Again, ai'm still not sure what your point is in all this, but if you're looking for solutions to the problem of getting power from a stationary bike withoit a power meter, an alternative for you may be to purchase a set of power meter pedals, such as Garmin Vectors, and mount those on your existing stationary bike. Those measure, obviously, force at the pedal, and you'd need an ANt+ enabled receiving unit to see the output, e.g. the Garmin Forerunner XT.