@tinselearedcow
To me the Caaaaaaaaaaardiff accent is very distinct and totally recognisable as a Welsh accent.
"I'll have a pint of daaaark in caaardiff aaaahms paaahk" as my old uni lecturer used to say
Your bf is wrong to say most people in Cardiff don't have Welsh accents - people who have been born and bred in Cardiff (particularly if their parents weren't necessarily professionals, as your bf said) often have the cardiff accent which sounds quite different from the traditional south wales valleys accent on gavin and stacey.
Perhaps your bf is just a bit limited with whom he socialises with - unlike most other welsh counties, Cardiff has a few private schools, so again if he's making the class/education difference that could be why people he knows might not have stronger 'welsh' accents, but again, def doesn't apply to everyone in Cardiff!
If you watch Rob Brydon on any of his comedy shows, his accent is still clearly welsh but far less exaggerated than when he is playing uncle Bryn.
OP would you get confused about people living in Knightsbridge or Chelsea who didn't sound like the cast of Eastenders? Or would you understand that there are multiple different 'London' and indeed 'English' accents?
Does your bf sound like Gareth Bale, for example? Brought up in Cardiff but doesn't have a strong Welsh accent, but try saying he's not Welsh!