Does renaming a display also rename its camera and microphone?

No. BetterDisplay's display name identifies the screen. Its camera and microphone are separate devices. Renaming a Studio Display in BetterDisplay does not rename the camera or microphone entries shown by a video-call app or website.

BetterDisplay does not provide a camera or microphone renaming feature. If your calling app offers its own device labels, configure those there. Otherwise, use that app's camera preview or microphone input meter to identify and select the intended device. A custom display name is not a reliable way to distinguish those entries.