I've seen many personal web pages of (US/UK/EU) academics that have a list of all the PhD, MSc/BSc or internship students they've supervised before. Is this allowed or does it break privacy of the students?
The governing federal law in the USA is FERPA - the Family Educational Rights and Privacy Act. Under FERPA a school may generally disclose what is identified as "directory information," things such as names, email addresses, photographs, etc. However the student has the right to request that their directory information be excluded from public disclosure, so it's possible that they are not covered here.
Note that, as in most cases, individual states may have their own privacy statutes.
If you are really concerned then the prudent thing to do, regardless of legal jurisdiction, would be to ask the student in an email whether they are OK with the posting of their name and potentially other information. The email provides a reasonable level of proof that you did your due diligence prior to posting student information.
US DoE guide to FERPA: https://www2.ed.gov/policy/gen/guid/fpco/ferpa/students.html
Many professors do this to establish the "academic family tree" of their lab. Most of the students are happy about it and proud. It only documents that you spent some time in that lab/group working on a topic, which I don't think violates your privacy. If you don't want that, you can contact the professor to get your name removed; shouldn't be a problem.
I don't know in the UK, but at least in Spain these data (at PhD level) are public. Not only student and supervisor, but also committee members, etc. Here an entry of the database: https://www.educacion.gob.es/teseo/mostrarRef.do?ref=1381779
I wouldn't consider that as a breach of privacy but actually good transparency.
This not only allowed but is a good practice too towards transparency, and is definitely not a breach of privacy.
Supervision is a publically known position, not secretive and the website is only making it known to a wider audience.
If the student's private emails or personal information of their private travel, etc were revealed on a public site, that would be a breech on the reasonable expectation of privacy.