I see registering your place of living is quite importand when dealing with local taxes. Ie when staying in one place for longer than some period of time, alos same information is used for social purposes, for example to assign your kids a place in kindergarten or school. It just so happens that this responsibility is assigned to police. By the way passport issuing is also i think is under police in LT. And yeah tourists who stay for long periods are handled with same subroutine (if i can say so). Stay at one place for prolonget period of time, please register.