So indeed, while ship is Swedish owned, being British flagged means we are the ones running around here.
I wouldn't make that assumption based on headlines - you don't know what diplomatic efforts are going on from the Swedish side.
I mean the "Iranian" vessel seized in Gibraltar isn't Iranian flagged, it uses a Panama flag but the press make reference to it being an Iranian vessel, seemingly the flag isn't relevant in that case.
I think you'll find that it makes for better headlines and is more relevant to the ongoing story as a whole to emphasise the UK flag with regards to the vessel seized by Iran. Yes the UK does have some obligations and is of course objecting but it is a Swedish ship and so I'd be very surprised if the Swedish government wasn't too despite not being the focus of newspaper headlines.