EDIT: See edit in my previous comment on how Bluetooth can do it. I believe that’d work with any device that can do bluetooth, including iPhones.

I suppose along an iPhone? I mean Apple does the whole ecosystem. And this isn’t really a technical limitation. Most phones have the audio stream connected to the processor. Theoretically they could forward it, or record it. But on Android, the often don’t seem to allow any of that, and Apple doesn’t allow third parties (like a Linux computer) to access “their” interfaces, so I don’t know if you can forward it to arbitrary computers either.

I mean there are solutions. Other people here outlined that. For example mimicking a bluetooth handset. You could solder a cable to attach to a computer’s AUX input. Or use a landline or different service to manage the calls whithin a PBX. But none of that is very easy to set up or proper forwarding. Maybe the best bet would be bluetooth.