What happens if session name is same on two different websites?

Question

I have a two diff. project on my XAMPP say it is Project1 and Project2.
When i login with Project1, i check authentication and if it is successful then stored session. The session name is $_SESSION['username'].
The above process is same with Project2.

now,to prevent direct access,i use this code(in both project):

if($_SESSION['username']=="")
{
  header("location:index.php");
}

so when i login with Project1, i am also access Project2(without login).
To prevent this, i know that if i create diff. session name for both project then it is solved.

The above thing is in my local server. so i can create diff. session name for my all project.

But suppose my site is online and what happen if my session name is match with diff. site?
There is a millions of websites and there is a possibility that my session name is match with another website's session name.Then this might be happen that some user access my website with another website(in same browser) and he might be access my site without login.

So what happen if session is same for two diff. website? Can user is access my website without login? If yes then what should i do to prevent it?

Thanks in advance.

UPDATE

according to @Let me see's answer there is a possibility that if two sites are running on the same server then they may share the data.
So suppose the server is sharing then what should i do to prevent it?

Answer 1

Sessions are (usually) stored using cookies, and cookies are domain-specific. So, it doesn't matter if google.com or evilhackerdomain.ru uses the same session name as your app; your cookies are only readable/usable by the domains you specify. Even in the unusual scenario that sessions are managed in some other way, it will be domain-specific.