Azure Data Studio has the mssql extension essentially bundled into it (and that's what makes ADS a primary provider of mssql functionality). You don't need to download it. It is already installed by default.
You were not able to see it in ADS because it is not an extension for ADS, it is for VS code, to provide mssql functionality.
VS Code and ADS are two different tools entirely.
I am mostly will use it for MS SQL related things. like writing queries and etc.
Hence, I wanted to install this extension :mssql
You can do it in ADS.