separate_chaining | Hash table with separate chaining layout | Hashing library

The other day I was reading that article on CodeProject

And I got hard times understanding a few points about the implementation of the .NET Dictionary (considering the implementation here without all the optimizations in .NET Core):

• Note: If will add more items than the maximum number in the table (i.e 7199369), the resize method will manually search the next prime number that is larger than twice the old size.

• Note: The reason that the sizes are being doubled while resizing the array is to make the inner-hash table operations to have asymptotic complexity. The prime numbers are being used to support double-hashing.

So I tried to remember my old CS classes back a decade ago with my good friend wikipedia:

• Open Addressing
• Separate Chaining
• Double Hashing

But I still don't really see how first it relates to double hashing (which is a collision resolution technique for open-addressed hash tables) except the fact that the Resize() method double of the entries based on the minimum prime number (taken based on the current/old size), and tbh I don't really see the benefits of "doubling" the size, "asymptotic complexity" (I guess that article meant O(n) when the underlying array (entries) is full and subject to resize).

First, If you double the size with or without using a prime, is it not really the same?

Second, to me, the .NET hash table use a separate chaining technique when it comes to collision resolution.

I guess I must have missed a few things and I would like to have someone who can shed the light on those two points.