I don’t really know

I don't think it exists, people just often get all the education they could ever want at Hogwarts or whichever wizarding school is closest to them. Then they go get a job of some sort, unless they are from a super wealthy pureblood family I guess.

The final two years at Hogwarts IS pretty much university, by the end of year 5 your are expected to know what career path you are heading down, and every course you choose to keep taking after that point is to help train you for said job, and in 7th year you even do a co-op with that profession for a little while for some hands-on training. After graduation some jobs will do additional training such as Aurors, but many Ministry positions just take you straight from Hogwarts because that’s what your last two years there were for.

J.K. Rowling has said it pretty much doesn't exist so that's why she didn't write an eight book.

Maybe noy in the UK but some countries may have wizarding elementary, collage, university

Maybe higher education is the traditional become apprentice to a great witch/wizard, and only through this method knowledge of the most powerful magic- like horcruxes- passed down. Wizarding world goes in for a lot of tradition. I mean, what is with quills? Hasn't anyone heard of pens?