All the BBC podcasts I've listened to are affiliated with a BBC Radio station, so I don't see why they can't come under that umbrella.
That's the point, until fairly recently BBC podcasts were just radio programmes either in full or highlights. Early on there was a rule that they couldn't put out on a podcast anything that hadn't been on air, which is why if you turned on Radio 1 in the early hours of the morning you'd hear the Chris Moyles Podcast as the extra content they included needed to have been on air.
Over the last year they've started doing original podcasts to varying success, the most well known and successful probably being 'Me You and The Big C' which got a lot of coverage recently when one it's creator and presenter, 5 Live's Rachel Bland, died. Another success is Fortunately with Fi and Jane, which might be of interest to members here as it often interviews newsreaders and other BBC names. They are only loosely connected with their respective networks because they are productions in their own right
They recently got a head of podcasting and are now actively commissioning them
It's long overdue really, I think there was a fear that the BBC and as a consequence the UK in general were falling behind in the podcast world - the BBC despite all their might and creativity could only produce cut downs of Scott Mills and the like. It's a bit telling that one of the worlds most popular podcasts is presented by a Radio 1 presenter (although I can't imagine the BBC commissioning Dad Wrote a Porno!)
They're still linked to the radio stations and shows though and even if they weren't the iPlayer has content not broadcast on TV anyway. I've no issue really though with separate branding for their audio apps - it's just "BBC Sounds" is so ****.
I preferred the internet when it had a sense of humour.