The Zoom app must place the entry into the Microphone & Camera sections in the Privacy & Security System Settings. Once the Zoom entry is within those locations, then you can control the setting.
Make sure the Zoom app is completely up to date with the most recent version available.
Do you still get the Zoom app to prompt you to allow access to the Microphone & Camera? If not, then try to test the camera & mic in Zoom to see if that will trigger another prompt. Try quitting the Zoom app completely, reboot the computer, then try launching Zoom again to see if you are prompted to allow access to the Mic & Camera.
Try creating another macOS user account. Log out of your current macOS user account & log into the new macOS user account. Try using Zoom to see if you are prompted to allow access to the Mic & Camera. See if it takes in this new user account. If it works, then there is something wrong in your main macOS user account. If it still does not work correctly even within a new macOS user account, then there is a system wide issue of some sort.
Are you using any anti-virus apps, cleaning/optimizer apps, or third party security software? Even though booting into Safe Mode should have disabled them, sometimes they may make a modification to macOS which is still in effect even when those apps are not running. If you are using those types of apps, then uninstall them by following the developer's instructions. Actually with an app like Zoom may not work properly in Safe Mode since part of Zoom may have been disabled.