Does using user-specific accumulative variables causes data leakage? Let's say I have a scenario in which my observational unit is a bill that was issued after a certain service was given and my goal is to predict if this bill is going to be paid or not. I have users in the system so I include user-variables like number of unpaid bills the user has, user's time in the service system (seniority) etc. I train on month 1 and test on month 2 (bills that were created in those months, respectively).

In the testing month I will have user-variable's count increased, so e.g. if during training user_1 had 100 days time in my system, of course that if there is a bill associated with him in the test month - his count of days will be higher. 

Is this accumulative nature of such variables is considered a data leakage between train and test sets (because part of the information that was used in training is being used, in a sense, in testing)?
If the data is available at prediction time, then it is not data leakeage.
In your specific example, historical user data should be used at prediction time to build a better model.
You do have to decide on how to do the train/test split. You can split by user, by time, or combination.