Hello BL2323,
To help us get a better understanding of the issue, we’d like to ask a few questions.
1. What happens when you try to connect that mouse to your old MacBook Air? Do you receive an error? If so, what does the error say?
2. You mentioned that the old MacBook Air shows other devices; just to confirm, are you able to connect other Bluetooth devices or accessories to that Mac without an issue?
The above information will help us plan how to proceed.
Please also check out the information outlined in the following link to ensure the correct process is being used to connect Bluetooth accessories and devices, so we can make sure there's no issue there: Connect a Bluetooth device with your Mac
Connect a Bluetooth device with your Mac
Connect your Mac with a Bluetooth keyboard, mouse, trackpad, headset, or other audio device.
1. Make sure the device is turned on and discoverable (see the device’s documentation for details).
2. On your Mac, choose Apple menu > System Preferences, then click Bluetooth.
3. Select the device in the list, then click Connect.
If asked, click Accept (or enter a series of numbers, then press Enter).
You need to connect your Mac with the device only once. The device remains connected until you remove it. To remove a device, Control-click its name.
You can also connect a Bluetooth device to your Mac by selecting Bluetooth status icon in the menu bar and selecting the device.
Thanks!