This is for Germany. Other countries differ, but the visa is a thing for all of them. Friend of his in Switzerland got his visa through quite fast. Germany definitely seemed the worst - see below.
It took far longer for a visa to come through than expected. DS applied for his at the end of August and visa came through on 18th December, by which time he was legally outside the 90 days you are allowed to be there. He got a temporary visa granted in October but that didn't allow him to re-enter the EU if he left, so it was a bit hairy just before Christmas when his permanent visa hadn't arrived.
There is nothing like the same support for students within universities, it's very much hands off and negotiating exam timetables, lectures etc was sometimes interesting. DS had an timetable clash which he was trying to sort and one of his lecturers told him to get it sorted at once or she'd fail him, but no one from the other department got back to him. He did sort it but it was nerve wracking. Exams had to be booked - told to go on to a website to see exam timetable to book them, but none of the foreign students could actually log in so they were all really panicked, then they were finally allowed on in the afternoon and it was fine.
He had to make an appointment at his university accommodation to be allowed in with the caretaker, and then he had to have an appointment when he left so the caretaker can agree the room was in the same state as when he arrived.
In Germany you have to put money into a blocked bank account out of which insurance, rent etc then come. As he was there for 6 months he had to put in €5500. Germans don't do things by email or phone - everything was an in-person meeting, whether to open a bank account or sort a sim card.
He couldn't work while he was there - and anyone's advice which is based on experiences prior to 2020, or children with EU passports, are, very sadly, no longer relevant apart from the British Council info. DS did lots of courses at his uni, actually ended up with more credits than he needed for his UK one, but that was fine. Being at uni meant he met lots of people. My niece did her year teaching English and she didn't come out as fluent as the students on university courses so worth considering.
We booked his flights as follows: return Club Class (two bags in the cabin, plus two 23kg bags in the hold). He took 2 hold cases and a rucksack so he could fit duvet, covers, pillows etc, along with winter clothes. He took those flights out in August, and back at the end of the whole stay. Christmas we booked (in September as it gets very expensive), as Economy return.
Can't think of anything else at the moment. Hope that helps.
Best experience of his life - has made incredible friends from all over the world.