live Kali Linux doesn't see MacBook Air internal drive

If you cannot see the SSD then the problem is that Kali Linux didn't include the drivers to support Apple's SSD.

You are much better off running Kali in a VM than trying to dual boot on a Mac. Only some Linux distros partially work on Macs (barely). The kernel might require patches, add-on kernel modules, etc. to support the built-in keyboard, trackpad, disk controller, etc. It can be an incredible challenge and then you could sleep the Mac and wake only to find the keyboard and trackpad stopped working, etc. The Kali distribution may not be as up-to-date as other mainstream distros in regards of hardware support outside of PCs.

I've never used Kali Linux so I don't recall what parent distribution Kali is based upon. Make sure you are using the most recent version of Kali Linux. I would expect Kali to be able to recognize the partitions, but Kali most likely won't be able to see the macOS file systems especially APFS file systems. You would definitely need to use Disk Utility to create a new partition for Kali to use. Kali will need to delete that partition so that Kali can divide up that space to create the partitions Kali requires. Don't use any third party OS to resize or split the partition of another operating systems boot volumes.

As @James Brickley mentions, not all Linux distributions work well with Apple hardware out of the box and may require some tweaks to even get Linux to boot after installation even if the installer booted fine. The only Linux distributions that I believe work well out of the box are Linux Mint and the Ubuntu flavors. While you should be able to get any Linux distribution to work on a Mac, that may require a lot of extra research & work on your part. If a Linux installer has a problem seeing the hardware, then you will need to look at that Linux distribution's installation instructions and forums for assistance to enable any special drivers (may not be possible for some distributions). Once Linux has been installed, then installing drivers is fairly easy if they are part of the Linux distribution's software repositories, or you may need to compile it from source code (not always so easy).

I never recommend anyone to dual boot any computer especially by using multiple partitions on the main internal boot drive. You are just asking for trouble. People always end up with a broken macOS and the inability to access their data due to the resizing of partitions which is always risky. People almost always choose the wrong partition sizes for each OS or they end up wanting to delete the extra OS which results in not being able to merge the unused space back into the main partition. This usually requires a clean install of macOS by first erasing the whole drive.

A better option if you need to install Linux on bare metal is by installing Linux to an external USB drive. As long as you install the Linux bootloader to the external drive, then there is minimal risk to the internal macOS boot drive.

The best option is to run Linux (or any other OS) in a Virtual Machine as @James Brickley suggested since there is minimal chance of damaging the host OS this way. This should always be your first choice unless there is a need for GPU hardware acceleration or full access to memory & CPU for optimal performance. Dual booting is always a pain and annoying, while launching an OS in a VM is much less disrupting.