private static async Task Foo()
{
    await Task.Yield();
    throw new OperationCanceledException();
}

private static async Task ShoudlyVariant2(Type exceptionType)
{
    try
    {
        var exception = await Should.ThrowAsync(Foo, exceptionType);
        Console.WriteLine("ShoudlyV2: " + exception.GetType());
    }
    catch(ShouldAssertException s) when (s.Message.Contains("but did not"))
    {
        Console.WriteLine("Variant 2 did not got " + exceptionType.Name);
    }
}

// manual variant
try
{
    await Foo();
}
catch (Exception x)
{
    Console.WriteLine("Manual:" + x.GetType());
}

// Shoudly variant 1
var exceptionV1 = await Should.ThrowAsync<Exception>(Foo);
Console.WriteLine("ShoudlyV1: " + exceptionV1.GetType());

// Shoudly variant 2
await ShoudlyVariant2(typeof(Exception));
await ShoudlyVariant2(typeof(TaskCanceledException));
await ShoudlyVariant2(typeof(OperationCanceledException));

Manual: System.OperationCanceledException
ShoudlyV1: System.Threading.Tasks.TaskCanceledException
Variant 2 did not got Exception
Variant 2 did not got TaskCanceledException
Variant 2 did not got OperationCanceledException