Thank you for your informative and descriptive answer, Arachne.
Sadly, no other brand of dvd wallet-cases is available in my country (Lithuania).
And I found the cause - the case is made very durable, and as such, the walls and sockets are made from, respectively, specific nylon and nylon.
But this specimen has too tight sleeves, so the dvd, that is inserted, is hard-pushed into the sleeve.
Finally I've thought of an idea - to untighten those sockets by roughly inserting three DVDs at once (i have badly-burned dvd's for such purposes).
Then the precious DVD can be inserted with much gentler force and the problem is solved for all future dvd's. Unless, of course, the nylon walls are so messed up that the nylon scratches DVDs instead of protecting them.
So the problem is kind of solved, unless future experience shows otherwise
Thank you all for your answers!