This is all good stuff.All rights are assigned on a territorial and platform basis. EPL, EFL and FA for their Cup competitions limit the rights for the clubs as part of the competition rules.
The club can only sell local broadcast rights to radio where there is a territorial provision in place, which is why you can get it on FM, DAB and Freeview as they all have a regional broadcast footprint. The BBC does not acquire the rights to broadcast this via the Sounds App or on its streaming server. The club cannot sell the TV rights, for example, they are managed collectively by the EPL, EFL or FA depending on the competition.
When the BBC buys rights for its Premier League package for BBC5Live, for example, it does acquire the right for the Sounds app but not for the streaming server, so you can listen in the Sounds App, but not in any other app that relays the BBC streams.
Similarly, when Sky Sports buy the TV rights to EPL games, it can only broadcast to UK subscribers. This means the EPL can then sell to territorial rights to various other international broadcasters for their territories.
One addition - streaming platforms, like Sky Sports, used to be allowed to stream their content, quite legally, to UK subscribers elsewhere in the European Union - for example, those on their holidays.
Unfortunately, when Britain left the EU, the UK broadcasters now no longer do this (they could have, but chose not to) and such access is now geo-blocked (which is frankly nonsense, but there you go).
EU law mandates that an EU subscriber of a streaming service can legally access their service wherever they are in the Union. When I travel for business, for example, I can access my Sky Sport DE and DAZN services quite legally wherever I am in the EU (and in a couple of other countries, too) which is nice.