Uploading a file that is over 10GB on cloud

See the section : If your files don't update to iCloud Drive in the article https://support.apple.com/en-in/HT203052

Read this https://support.apple.com/en-in/HT201104

What types of files can I store in iCloud Drive?

You can store any type of file in iCloud Drive, as long as it's 50GB or less in size and you don't exceed your iCloud storage limit. That means you can keep all of your work documents, school projects, presentations, and more up to date across all of your devices. Learn more about managing your iCloud storage.

You shouldn't store app folders, libraries, or .tmp files in iCloud Drive.

Note : in your question you mentioned that 2 TB storage is the maximum limit used , then again there is shortage of space , have you exceeded the 2TB storage also , clarify .

See some articles to reduce the data https://support.apple.com/en-in/HT204247

https://support.apple.com/en-in/guide/macbook-air/apdb468896e0/2019/mac/10.14.5

https://support.apple.com/en-in/HT206996

https://support.apple.com/en-in/guide/system-information/syspf5a64aa6/mac

As per this article https://support.apple.com/en-in/guide/mac-help/mchl027f1d66/mac

If you don’t have permission to view or work with files on a storage device

• If you are not the administrator of the Mac, ask the administrator to give you access to the files you need.
• If you are the administrator of the Mac, do one of the following.
  • Authenticate as an administrator user: Depending on how you’re attempting to access the file, you may need to authenticate as an administrator user.
  • Change Sharing & Permissions settings for the file: For instructions, see Use file sharing.

Now search the file that you copied in internal hard disk to some location , click on finder > Go > computer > Macintosh HD

See https://support.apple.com/en-in/guide/mac-help/mh15155/mac

https://support.apple.com/en-in/guide/mac-help/mchlp1008/mac

If you find the file right click on it and use get info , what are the permissions set up change it to read and write https://support.apple.com/en-in/guide/mac-help/mchlp1203/mac

Also you can compress the file https://support.apple.com/en-in/guide/mac-help/mchlp2528/mac

Which system version is installed on your Mac? On newer system versions the file size limit for iCloud is 50GB, but on older system versions it has been much lower.

If your system does not let you drag the large file to iCloud Drive, try to upload it using the web interface at www.icloud.com

But if your Mac is running low on storage, it may not be a good idea to move such a large file to iCloud. You will need the 50 GB local storage, if you ever want to use it again, because it requires a download from iCloud. And working with such a large off-line file will be very slow. Consider to move the really large files to an external volume instead, where you can access them directly, without needing to download them from iCloud.

It is weird. I copied it into iCloud Drive but then my mac started showing me notifications that I have a low storage space

That is normal. Files you are keeping on iCloud Drive are having shadow copies on your Mac, so you can still work with them, when your mac is off-line. If you enable the option "Optimize Mac Storage", iCloud will remove the shadow copies for the files you have not used in a long time, but only, if your mac is low on storage. Did you already enable "Optimize Mac Storage" in the System Preferences > AppleID > iCloud?

What is version of Mac , do you use any vpn services / anti virus , softwares its query .