Thank you all for the replies. The silver coloured screws (or whatever they are) all look similar and seem to be made of metal (it's difficult to see on the pictures). Since the "screw" looks like they're made of metal and the casing definitely being made of plastic, I'm afraid to use a drill bit.
Torx security screws actually were my main suspects. The edges don't look perfectly round but a bit jagged indeed. A bit holder won't fit into the narrow openings where the screws are. So I suppose I would need a "regular" screw driver of the right type? I have no idea which size I would need. Otherwise I would already have ordered a torx screw driver. I haven't tried screwdrivers that are used for mobile phones. (I have no idea which type they are.)
It puzzles me to find those screws in a system that was built somewhere in the late 80's or early 90's and isn't supposed to come with anything but Phillips head screws. I haven't found a single piece of information about original model NES systems using anything else. I was actually quite afraid when I saw this first. ("Some lunatic must have opened it and it won't work at all.") It also could have been a Blue Monday somewhere in Japan when they built the unit and someone decided to use a different set of screws and the unit being left unopened ever since. But the system worked perfectly on first try, which really surprised me. Old cartridge based systems require cleaning of the cartridge slot (and every cartridge that's put into it) and usually won't work on first try. The NES is even more prone to reading errors because of the extra stages of potential problems due to its design (the cartridge slot not being connecting directly to the main board and its pins that are bent when the cartridge is in the system, which will eventually lead to pins getting bent out of place).