Is that message unique for updates with a minimum stability?

I think its a global error message. I see no error msg here https://github.com/joomla/joomla-cms/blob/1e3eb590cda5e8b06f6f4c5feae7fd2d0e45528c/libraries/joomla/updater/adapters/extension.php#L131

But the next question is also, why I can see the update. I set the option as stable.

As far as I know (and be aware I might be wrong), the minimum stability feature refers to the process of discovering updates (the findUpdates method).

The error you refer to (with the language string COM_INSTALLER_INVALID_EXTENSION_UPDATE), is actually triggered in the InstallerModelUpdate::install function (which has the comment "Handles the actual update installation.").

Rather than change the language string (which I dont think we can as it is generic) isnt it better to resolve

But the next question is also, why I can see the update. I set the option as stable.

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_{This comment was created with the J!Tracker Application at issues.joomla.org/joomla-cms/6920.}

Hi you created this issue sometime ago but have not provided any code for people to evaluate. As no one else has shown any interest in providing the code and you have not then I am closing this issue at this time. If code is provided (a pull request) it can always be re-examined.