Disk not showing in recovery mode

Linux Mint cannot read for macOS storage device formats. The device needs to be erased. Select the device, and erase it. The device in this context the the one shown with an immutable name such as "Apple SSD storage media" or words to that effect.

If you want to have macOS installed, then you need to create a partition for it first. macOS does not recognize or understand Linux file systems so Disk Utility can not modify the disk with them. Same sort of also applies to Linux & Windows.

With Big Sur, you would need to first reveal the physical drive within Disk Utility since the physical drives are hidden by default. Within Disk Utility click "View" and select "Show All Devices" so that the physical drives appear on the left pane of Disk Utility. You will need to select the physical drive and Erase the physical drive as GUID partition and APFS (top option) if you wish to install Big Sur.

If you are booting from macOS 10.8.5 Mountain Lion, then you will need to select the physical drive and create 1 Partition using the MacOS Extended (Journaled) file system. See the instructions in the following article:

https://eshop.macsales.com/tech_center/formatting/Mac_Formatting_6-10_R3.pdf

FYI, Both of these options will completely destroy all existing partitions on that SSD along with all their data, so make sure you have a good backup of that data first.

Also, if you were trying to dual boot macOS & Linux, or macOS & Windows, then you need to use macOS to resize the existing partition to allow the creation of a new partition for Linux to utilize, or use Bootcamp Assistant to install Windows (Bootcamp Assistant takes care of repartitioning the drive for you).

<< the disk is not showing [...] neither in diskutil list >>

... if you are not already making progress, ¿what DOES show in diskutil list?

Select the text in the Terminal window and copy and paste here. Here is mine from Ventura, so it will be VERY different:

diskutil list

/dev/disk0 (internal):

#: TYPE NAME SIZE IDENTIFIER

0: GUID_partition_scheme 1.0 TB disk0

1: Apple_APFS_ISC ⁨⁩ 524.3 MB disk0s1

2: Apple_APFS ⁨Container disk3⁩ 994.7 GB disk0s2

3: Apple_APFS_Recovery ⁨⁩ 5.4 GB disk0s3

/dev/disk3 (synthesized):

#: TYPE NAME SIZE IDENTIFIER

0: APFS Container Scheme - +994.7 GB disk3

Physical Store disk0s2

1: APFS Volume ⁨Macintosh HD⁩ 15.4 GB disk3s1

2: APFS Snapshot ⁨com.apple.os.update-...⁩ 15.4 GB disk3s1s1

3: APFS Volume ⁨Preboot⁩ 623.3 MB disk3s2

4: APFS Volume ⁨Recovery⁩ 822.7 MB disk3s3

5: APFS Volume ⁨Data⁩ 440.1 GB disk3s5

6: APFS Volume ⁨VM⁩ 20.5 KB disk3s6

"Disk not showing in recovery mode: Hello, I have a MacBook Air 2013 13 inch Retina display, and I’ve been installing windows and different Linux distros in it and after I installed Linux mint on it I tried to install macOS 10.8.5 but the disk is not showing in disk utility neither in diskutil list, i’ve tried to use Big Sur recovery screen but the disk still doesn’t show there either. I’ve reset smc and vram but it still doesn’t show it still."

-------

Installing the macOS 10.8.5 on a MacBook Air:

To install the macOS 10.8.5, you are limited to the year of release of the model of your MacBook Air. So, you'd need a much older Mac to do so, and one with an Intel processor. MacBook Air (Late 2008 or newer) are the models that will install this. Any Mac that shipped with Mac OS X Mavericks or later will not be compatible with this installer. The installer: Mac OS X Mountain Lion Installer - Apple Support

In depth...

Installing the macOS:

Determining if macOS Lion will Install:

Verifying Compatibility:

How to Verify:

I. Reveal the Serial Number in Terminal:

1. Boot: in Recovery Mode
2. Go to: Utilities Menu (at top of the screen)
3. Click: Terminal
4. Use this Command:

system_profiler SPHardwareDataType

then...

II. Enter the Serial Number Here:

View Coverage - AppleCare & Warranty - Apple

then...

III. See if It Can Be Installed:

Start Here: Learn About Compatibility and how to Identify your Specific Mac Model - Apple Support

<< this is the link with the video of my diskutil list it doesn’t show my internal disk at all. >>

The first item is a GUID partition that is 2.1 GB in size. That is the ROM-based disk image that contains MacOS base system, which is a stripped down minimal MacOS that runs Recovery and its Utilities. Recovery can work without a working drive because it creates more than 25 RAM disks, which are shown in that display as well.

But since no possible boot drive appears in that list, in ANY format, that suggests the boot drive has FAILED

This older Mac does not have a T2 chip, so you could connect and install onto an External drive to make progress.