Interesting. I graduated from the _other_ UW (University of Warsaw) and our uni has course-swapping capability built into the University Study Service System (USOS)[1].
FYI public university education is fully government-funded in Poland (i.e. it is free for students).
Isn't USOS like nightmare of a system? I've heard of stories that people had to graduate after time, because they failed a subject at first year (!) and they had no chance to sign up for the course in the following years at USOS because someone was always faster.
USOS is just a tool and it's mostly a matter of how it's configured (for example MIMUW, my faculty, doesn't use first-come, first-serve registration at all) and how various policies are enforced.
Stories about students failing to sign up for a previously failed course are mostly urban legends, I believe. The dean's office has the power to override anything that's going on in USOS and they can manually register you for a subject if it's needed for you to graduate.
I'm from New Zealand, and this kind of stuff isn't a thing where I went (as far as I'm aware). Course enrolments open at some point (everything all at once), and you just log in and fill it out at some point over the multiple months between it opening and the due date for completing it. Some programs have limited admission (with their associated papers being restricted to those enrolled in that program), but limited space at the paper level (as opposed to the course level) and rushing to submit your paper selection just isn't a thing (as far as I'm aware).
FYI public university education is fully government-funded in Poland (i.e. it is free for students).
1 - https://usosweb.mimuw.edu.pl/kontroler.php?_action=news%2Fde...