Hello everyone! I'm on record at a polyclinic and, for my own peace of…

Hello everyone! I'm on record at a polyclinic and, for my own peace of mind, I also registered at a private clinic. I have an exchange card in hand from the polyclinic. The private clinic won't give me one. The doctor says "why? Everything is here. Why re-write previous analyses etc. into a new card if you have this one? Here any doctor can keep records". BUT at the polyclinic they once made a remark "why did he write that another doctor?". And then the question: should I get a card at the private clinic or is this exchange card the same and you can go to the clinics where you are registered with it?