I was having the same issue with the error "Microsoft.Data.SqlClient is not supported on this platform." Same thing, upgrade from a 10.8.1 version using the ArcObjects SDK to using the ArcGIS Enterprise SDK 11.1. The SOE uses the SQL client to get some of the service data back from the source DB.
The SOE compiled successfully and even registered successfully as an extension in ArcGIS Server Manager. As noted, I got the "Microsoft.Data.SqlClient is not supported on this platform." error once I tried to assign it as a capability to Map service.
We requested support from ESRI on the issue and it has been identified as a bug (BUG-000158547) and should be resolved in a future release. However, I was also provided with a workaround that is working so far.
The compiled SOE found in bin folder that is copied to the server can be unpacked and updated - as in, rename the extension to ".zip" and you can expose the install files that are packaged up in the SOE. The compile folder also contains a folder of runtimes that has an updated version of the Microsoft.Data.SqlClient.dll, (bin\Debug\net6.0\runtimes\win\lib\net6.0) and I was able to copy that dll into the SOE Install directory, rename the file back to the SOE extension and update the SOE on the server. Then worked as a capability with the Map service I assigned it to without errors. Hoping this helps!