... and you are right about the Lighting vs. USB-C ports used on iPhones.
What you’re seeing is actually normal behavior on Lightning-based iPhones, and it usually isn’t a problem. iOS often identifies any analog TRRS headset-style input (mic + headphone combo) generically as a “Headset” or “Headphones” device rather than by the microphone’s brand or model. The Sennheiser MKE 400 connects via a TRRS analog interface when using the 3.5 mm path through Apple’s Lightning-to-3.5 mm adapter, so the system doesn’t have a digital handshake that exposes a specific device name. Apps like Blackmagic Camera simply report whatever label iOS provides.
This is different from USB audio devices or newer USB-C iPhones, where the mic may enumerate as a distinct USB audio interface with a recognizable name. On Lightning models, the analog audio is converted by the Apple dongle’s internal DAC/ADC and presented as a generic headset input. Because of that, seeing “Headphones” in the Blackmagic Camera app is expected and does not indicate a malfunction — especially if levels respond correctly and audio records cleanly.