At Warwick Mathematics Institute in the UK they do indeed have such a seminar, called the Postgraduate Seminar. It's organised by PhD students for PhD students only and the talks are given by PhD students.

In answer to your question, students can either present updates on research they are working on or they give a survey of some existing work (see, for example, my talk on noncommutative geometry and conformal field theories).

https://warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/postgraduate/