How to get two string variables passed from the base .cshtml page into the .cs page and the partial view?

You can pass a value from JS code to server-side code using query strings. In your case:

.fetch('testpage/?handler=TestPartial&' + 
  new URLSearchParams({ testing, testing2 }))

This will request the URL testpage/?handler=TestPartial&testing=xxx&testing2=yyy

You need to declare those parameters in your server-side method to parse and receive them:

public PartialViewResult OnGetTestPartial(string testing, string testing2)
{
  ...
}

Your parameters will be passed to your method this way. To show these parameters in the view, you need to pass it as a model to the view:

public class TestPartialViewModel
{
  public string Testing { get; set; }
  public string Testing2 { get; set; }
}

public partial ViewResult OnGetTestPartial(string testing, string testing2)
{
  return Partial("Partials/TestPartial", new TestPartialViewModel 
  { 
    testing, 
    testing2
  });
}

and declare your viewmodel class as the model for your view at the top of your view:

@model TestPartialViewModel

The value of testing is @Model.Testing
The value of testing2 is @Model.Testing2