Update: This seems to be an initial settings issue coming pre-set with the board. Once started for the first time It did not work properly generating the issue described (only one character per row was showing). This problem persisted via usb while trying to print text from the arduino's serial monitor as well as from LCD Smartie.
Then I connected it with the direct cable to arduino, it was printing from the test sketch as it supposed to. Then I downloaded a program called Realterm. It was connecting and printing text as it supposed to.
After this point these programs probably changed something in the backpack's conenction settings, because after these it works now as it supposed to even with arduino's serial monitor.
After attaching the brand new backpack to the lcd, first started up by connecting it to the computer via usb. Then installed the driver (the inf file from downloads) and straight away tried via the arduino's serial monitor: where it generated the issue. Then tried with LCD smartie where it had the same issue. Finally I disconnected it from the PC and connected it to an arduino via the JST cable provided with the display. Downloaded the sample sketch and it did work as it supposed to. At this point I did not check if the problem is otherwise fixed already so maybe this step already solved the issue.
I suspect the problem might have been at the lcd layout settings. It seemed like it has been set up not as a 16x2 but a 1x unknown display and since the sample sketch has it specified as 16x2 it works since then.
To check this I tested it as connecting the JST cable to the arduino and modified the sample sketch as it changes the LCD settings to 1x16 (see below)
lcd.write(0xFE); lcd.write(0xD1); lcd.write(1); // 1 columns (supposed to be 16) lcd.write(16); // 16 rows (supposed to be 2)
This change generates the exact same issue I had in the first place. I assume during the testing of the board after manufacturing the factory display settings remained in the board as 1 x whatever.