What site or app are you trying to use the camera for? Can you do it in Chrome?

You are right, it doesn’t have the same behaviour on all sites. It doesn’t work on Nextcloud Talk.

Is it something that the website controls or is it a setting in my browser that is specific to a site?

I was having the same issue and looked into it. A website using the `getUserMedia` API can force a specific device to be requested, which removes the ability to select a device from the permission prompt.

For example:

   // Forces the permission prompt for a specific device
   var id = (await navigator.mediaDevices.enumerateDevices())[0].deviceId;
   await navigator.mediaDevices.getUserMedia({ video: { deviceId: { exact: id } } });
   // Allows you to pick a specific device
   await navigator.mediaDevices.getUserMedia({ video: true });

I wrote up a userscript that defeats this. You can load it with any userscript manager (such as tampermonkey). May also run into issues with CSPs on some sites, copy-and-pasting it into devtools also works.

 // ==UserScript==
 // @name         Force allow camera choice
 // @version      1.0
 // @author       UniQMG
 // @match        *://*/*
 // @run-at      document-start
 // @grant       unsafeWindow
 // ==/UserScript==

 (() => {
     let original = window.navigator.mediaDevices.getUserMedia;
     original = original.bind(window.navigator.mediaDevices);
     window.navigator.mediaDevices.getUserMedia = (constraints) => {
         for (let devtype of Object.values(constraints))
             for (let key of Object.keys(devtype))
                 if (devtype[key].exact)
                     devtype[key] = devtype[key].exact;
         console.log("Fixed device request", constraints);
         original(constraints);
     }
 })();