Updating from catalina to Big sur- Macbook Air

Are you trying System Preferences> Software Update.

Always best when downloading from Apples Servers to be connected to the internet

via cable, if you can only use WiFi then move your mac as close to your router as possible.

Do not use the mac while it is downloading or any other devices that share the same internet connection.

Open your Safari browser and click on this link,

How to upgrade to macOS Big Sur - Apple Support

Do you have enough free space to install Big Sur, you need at least 35.5GBs of free space.

Suggest - restart Router and then Shutdown Computer and connect via Ethernet connection. Restart computer in Safe Mode - Shift - Key immediately at startup. It does a Repair Disk, clears cache files and load only Apple Software - load slowly - Normal. Once in Open Apple Apps Store and find Big Sur and download. Try not to let computer go to sleep mode or Screen Save to activate. Be patient as one knows it Big Sur is 12.18 GB size. There are often periods where it seems to pause the download , normal be patient as countdown clock may say 15 minutes but in Real Time could be 30 - 45 minutes. This can and will happen several time in the process. If all goes well. it will successfully download and attempt to Launch. Do NOT allow it to start - Quit the Installer. Check Applications Folder for new App called Install Bug Sur. Make a copy to External Drive - future usage. Now, restart again in Safe Mode - then launch the Installer.

Click on the  top let of your screen and click on About This Mac, post a screenshot o what you see.

Open Disk Utility from Applications> Utilities click view in the menubar (if you are running High Sierra or later)

and select Show All Devices. Post a screenshot of what you see.

So that checks out. Have you booted to your Recovery HD and then run Disk Utility First Aid

on the Disk and all its Volumes. So start with Apple SSD, then the Container disk, and then both Volumes

Mac13.6 and Mac 13.6 - Data.

Just wondering if the name Mac13.6 might be the issue, not sure if the OS likes a '.' in the SSDs name.

Have you restarted your mac in-between attempts.

I wonder if there is a problem with the Apple Servers, a few other people are posting the same issue,

perhaps try later in the day.

This is weird - in the supplied Pic it is Reading " This Update can not be installed on this computer ". By virtual of update specifically implies Big Sur is already installed. If the wording said the UpGrade could not be . . . - that would be a different matter.

Are you sure Big Sur was not already installed ?

What about the link I posted.

Something at your end then. Have a think and start at the very beginning.

By Apple Definition - Update is an update to existing macOS - example Catalina 10.15.7 to Catalina 10.15.7 Supplemental 2. Wherein an Upgrade is moving from Catalina to Big Sur.