Xunit create new instance of Test class for every new Test ( using WebDriver and C#)

While I don't know Selenium, I do know that xUnit.net creates a new instance of your test class for every test method, so that probably explains why you are seeing the behaviour you're reporting: the driver field is initialized anew for each test method, because the constructor is invoked every time.

In order to reuse a single FirefoxDriver instance, you can use xUnit.net's IUseFixture<T> interface:

public class Class1 : IUseFixture<FirefoxDriver>
{
    private FirefoxDriver driver;

    public void SetFixture(FirefoxDriver data)
    {
        driver = data;
    }

    [Fact]
    public void Test()
    {
        driver.Navigate().GoToUrl("http://google.com");
        driver.FindElementById("gbqfq").SendKeys("Testing");
    }

    [Fact]
    public void Test2()
    {
        driver.Navigate().GoToUrl("http://google.com");
        driver.FindElementById("gbqfq").SendKeys("Testing again");
    }    
}

after some investigation able to find the solution here it is and also updated FirefoxDriver to IWebDriver::

   public class SampleFixture : IDisposable
   {
    private IWebDriver driver;
    public SampleFixture()
    {
        driver = new FirefoxDriver();
        Console.WriteLine("SampleFixture constructor called");

    }

    public IWebDriver InitiateDriver()
    {
        return driver;
    }

    public void Dispose()
    {
       // driver.Close();
        driver.Quit();
        Console.WriteLine("Disposing Fixture");
    }
}

public class Class1 : IUseFixture<SampleFixture>
{
    private IWebDriver driver;

    public void SetFixture(SampleFixture data)
    {
        driver = data.InitiateDriver();
    }

    [Fact]
    public void Test()
    {
        driver.Navigate().GoToUrl("http://google.com");
        driver.FindElement(By.Id("gbqfq")).SendKeys("Testing");
    }

    [Fact]
    public void Test2()
    {
        driver.Navigate().GoToUrl("http://google.com");
        driver.FindElement(By.Id("gbqfq")).SendKeys("Testing again");
    }
}