Phd interview travel paid by interviewer - Is it bad form to ask to stay for the weekend?
If you make it completely clear at the time of asking that you would cover all additional expenses, then staying for a day or two extra would be regarded as perfectly normal and reasonable. (Academics travelling to conferences and workshops do this sort of thing all the time.)

Indeed, it may even be regarded as a positive sign: if you do join their PhD programme, then you'll be living in the city for several years, so the fact that you want to get to know the place in advance demonstrates that you're serious about your application.

My guess is that the somewhat intimidating wording "expected ... to depart on Friday" is intended to make it clear that you can't depart before Friday.