Sorting multiple arrays, based on the first array

I think the more appropriate option will be to create a new class with all different 6 kind of properties

public class myClass
{
    public DateTime date{get;set;}
    public string name{get;set;}
    //....
}

Then create a single array/list of that class.

public List<myClass> arrData;

Now you can sort that array based on any of your desired property and it will keep the order as per your requirements

arrData.OrderBy(x => x.name)

You can replace x.name with any of your myClass property.

This approach makes your code clean and easy to manage as well.

It looks like LINQ uses the quick sort algorithm for the OrderBy method (see previous StackOverflow question).
Something like this should take care of it for you:

DateTime[] datesOfBirth = new DateTime[] { new DateTime(1955, 10, 28), new DateTime(1955, 2, 24) };
String[] firstNames = new String[] { "William", "Steve" };
String[] lastNames = new String[] { "Gates", "Jobs" };

var people = 
    datesOfBirth
    .Select((_, i) => new 
        { 
            DateOfBirth = datesOfBirth[i],
            FirstName = firstNames[i],
            LastName = lastNames[i]
        })
    .OrderBy(x => x.DateOfBirth)
    .ToArray();

Likely you have a setup something like this:

DateTime [] dates;
string [] names;
int [] ids;
//...etc

Consider instead to condense your data into a single object, then have a single array of that object:

public class MyObject 
{
    DateTime date { get; set; }
    string name { get; set; }
    int id { get; set; }
}

Now you will only have 1 array:

MyObject [] objects;

And you can sort the whole collection of objects by their date:

objects.Sort((a, b) => a.date.CompareTo(b.date));

Also consider using Lists rather than straight arrays, as the use case for using vanilla arrays in c# is very small:

List<MyObject> objects;