Should the observation window include the same dates for all the users? I'm building a churn model that is trained on  users' historic data(observation window).
For this example, let's say that I want to train the model on the last week of use of each user.
Is it fine that the weeks of each window will be different according to the last week of use for each user? For example the next figure:

Are there are any advantages if the observation window starts and ends on the same dates for all the users?

The answer to this question is data-dependent.  The theoretical question is whether your data is stationary, i.e. does the data have the same statistical properties, regardless of when samples are taken.  The practical question is whether the data is close enough to stationary for your particular application, because perfectly stationary data is very rarely to never observed in commercial data.
If the data were perfectly stationary, then it wouldn't matter; samples taken from any one day will have the same properties as samples taken from different days.  Imagine the following urn-problem; if every day, a new urn is generated containing 100,000 red balls and 100,000 green balls, the sampling distribution doesn't vary (much) when 100 balls are drawn from the final urn versus when 10 balls are drawn from each from 10 consecutive urns.  However, this obviously changes, if there is a trend in the ratio of red to green in the delivered urns.
The safest thing to do is to define a few very basic statistics that describe what is occurring in your time-windows, something as simple as number of transactions in the window, time-between the last two transactions, etc. and then plot their value for a random sample of your data versus time.  If there is no trend, then you are OK using either method for a model of general predictability.
If there is a trend, then you probability want to use the most recent data, if you have enough data to make that an option.
