Several years ago, I found this course schedule from Cambridge. I also found some synopses from Oxford here$[1]$. They have been very useful because they contain complete course guides, for example (Cambridge):
Notice that it is very detailed, it has the total number of lectures, the number inside brackets is the (approximate) number of lectures to finish the contents in each paragraph. There are also very good book recomendations with a cross to mark some book which is particularly fit for the course.
Aside from that, the only other material I found that looks similar is Garrity's All the Mathematics You Missed: But Need to Know for Graduate School, it doesn't have all the details but it has a nice conversation about the subjects. I also found the following website: How to Become a Pure Mathematician (or Statistician).
I found these guides to be extremely useful.
Do you know more universities that provide such useful syllabi/synopses/schedules? I have been looking for some time, but it seems that Oxford/Cambridge is really unique with respect to this: Some of them just give the name of one book and doesn't detail it, other may have something like this but it's restricted for official students.
Do you know other books/websites/etc such as the ones I mentioned?
$[1]:$ I am a bit confused by the bureaucratic usage of the words "syllabus", "synopses", "schedules", etc. When I first searched for it, the name of the document was "syllabus", now it is "schedule".
As you said, these are usually restricted to enrolled students, or - more commonly - they don't seem to exist at all. If anything, it's not clear to me why Cambridge produces these, at least for Part IA courses, since I'm fairly sure most e.g. introductory analysis courses will cover roughly the same material; these might be useful for the course lecturers, or perhaps for students who choose to avoid the lectures / lecture notes entirely while learning, but not much else.
Are you aware that many Cambridge lecturers actually provide their own notes for free online? These are variable in quality, and not always easy to find, but they exist and are often very good. One that sticks in my mind is Keith Carne's Geometry and Groups notes (PDF): not only does his contents page give a breakdown of what's covered in each lecture, but it even gives a breakdown of what (almost) every individual theorem means!
You might find a few things on MIT OpenCourseWare - e.g. reading off the titles here.
Also see my note at the end of this answer.
You are right to call it bureaucratic. There is no standardised usage.
Here's a question from me to you: why do you want these? Specifically, why do you want guides produced by universities? After all, this is the purpose that textbooks are designed to serve - and textbooks are written by the same people who teach at universities! So, by avoiding textbooks, it seems to me that you're missing out on quite a lot.
"They're expensive" won't wash, either: Amazon will usually give free previews of the contents pages of at least some modern textbooks. For instance, here is Burkill's Analysis book - the one with a cross next to it in your picture. The contents pages are freely viewable, and about as detailed as the Cambridge syllabus you've posted above. (This is hardly surprising, given that Burkill was at Cambridge when he wrote it.)
Similarly, many universities don't produce syllabi precisely because their lecturers will be working from an easily available textbook. This textbook will sometimes even be written by one of the lecturers, but will almost always be written by a trusted lecturer at some university. It will usually have developed out of a course that the lecturer has taught in the past, too.