I don't think it can.
All the data regarding walking distance, burnt calories, etc. is calculated based on the data you enter in health on your phone (your age, height, weight, etc).
And there is no multi-user support there.
So when all this is set to you, the results for your wife would be all wrong by a margin.
The only way I would see is to un-pair and re-pair the watch with your iPhone and your wife's each and every time you switch the watch, which is a lengthy task. As far as I know it's not possible to pair a watch to two iPhones at the same time (only several watches to the same phone).
Maybe there are some third party apps that support multi-user, but I highly doubt that the health data displayed by the watch would be very accurate then.