Microsoft Bluetooth Designer Mouse

What macOS version is the MacBook Air running?

It appears the Mouse is compatible with up to macOS 10.10 but nothing more recent is listed, which may mean it cannot work with newer versions of macOS.

If the MacBook Air cannot see the mouse in the bluetooth settings there's not much do be done until it can see it.

There isn't anything special that needs to be done on a Mac to see a bluetooth mouse:

macOS : Pair with a Bluetooth device

Some more possible solutions here: Microsoft Designer Bluetooth Mouse is not detected in 10.10.3

and here: Can't sync Microsoft Designer Bluetooth Mouse to my Macbook

However, it appears the mouse is simply not very compatible with a Mac as others there report lots of issues getting it to pair.

Not recognized by what exactly?

Have you followed Microsoft's instructions to set the mouse to pairing mode?

Are you sure the mouse is compatible with whatever you are trying to use it with?

Microsoft Accessories Support - Microsoft.com

https://www.microsoft.com/accessories/en-us/products/mice/designer-bluetooth-mou se/7n5-00001