As noted in the title my session is not created, is it possible that it is the fact that I am a subdomain of a domain already having a session?

Here is my php code:

if(isset($_POST['login']))
{
    global $conn;

    $id = trim ($_POST ['identifiant']);

    $pass = trim ($_POST ['pass']);
    $pass = strip_tags ($pass);
    $pass = htmlspecialchars ($pass);
    $password = (hash ('sha1', $pass));

    $query = $conn->query("SELECT id, identifiant, password FROM tablename WHERE identifiant='$id'");
    $data = $query->fetch(PDO::FETCH_ASSOC);
    $count = $query->rowCount();

    if(($count == 1) && ($data['password']==$password))
    {
        session_start();
        $_SESSION['user'] = $id;
        header("Location: home.php");
    }
    else
    {
        echo 'Erreur';
    }
}

Here is my html code:

<form class="form-login" method="post" wfd-id="5">
     <center><img src="assets/img/logo.png" class="form-login-heading"></center>
     <div class="form-group">
         <input type="text" name="identifiant" class="form-control" placeholder="Identifiant" style="text-align: center;" maxlength="6"; required="">
     </div>

     <div class="form-group">
         <input type="password" name="pass" class="form-control" placeholder="Mot de passe" required="" style="text-align: center;">
     </div>

     <div class="form-group">
         <button type="submit" name="login" class="btn btn-login btn-block">Connexion</button> 
     </div>
</form>

home.php:

if(!isset($_SESSION['user'])){
   //header('Location: index.php');
    echo 'Session:'.$_SESSION['user'];
}

I see a few problems with your code, one of which will be responsible ... probably the last one.

problem #1: security

You're open to sql injections. trim doesn't prevent that. please learn about prepared statements. I cannot stress this enough, especially with a login form.

Also please use password_hash to create the hash for storing the value in the database and password_verify to verify the password you get is the right one. If your user table is compromised (which is likely, given you have an sql injection), a rainbow table will give easy (easier?) access to your user's accounts.

problem #2: weird assumptions

fetch plus rowCount. You absolutely shouldn't need both. What are your assumptions?

If you think that there might be more than 1 result row, then your database schema is flawed. The field should be marked UNIQUE. If you select on a unique field, there can only ever be no or one result. (also identificant suggests there should ever be at most 1 so unique is appropriate)

If you think that there might be no result row, then your fetch will just return null/false if there's no row, and you check for that.

also the php docs state, that calling rowCount on a select statement may not return what you want and you should make a select COUNT(*) query instead (there's also a comment re-stating that the only way to get the count after the select statement is to call a separate select count(*) ... statement or to count what fetch returns - essentially).

problem #3: redirect and not returning?

This is theoretical only. After header you should end the request by returning/exiting before you generate more output and make the header invalid. (Redirect + output -> output + error message (possibly))

problem #4: isset confusion

if(!isset($_SESSION['user'])){
   //header('Location: index.php');
    echo 'Session:'.$_SESSION['user'];
}

this will only output something, if no user is stored in the $_SESSION. either remove the ! or replace isset with empty