Wordpress multi-user, custom database, potential concurrency problems when using global wpdb

I am developing a Wordpress site, which will handle user registration, friends, teams and soccer tournament brackets.

On this site I have the wordpress core database and a few custom non-wordpress tables, that I have made myself.

I would like to use the Wordpress global $wpdb variable to make CRUD requests on my custom tables.

Question:

Will I run into concurrency problems with the $wpdb variable, when multiple users use it at the same time?

Example:

There are 10 users online, each of whom makes an INSERT call to the sql_tournament table. The INSERT command on $wpdb, when executed successfully, fills in the $wpdb->insert_id variable. Will each users INSERT request change the $wpdb->insert_id variable for everyone (aka. globally) or will everyone have a separate $wpdb instance, making it safe to use, because I can safely assume that when retrieveing the $wpdb->insert_id variable, I will get ID of the table that was inserted by the same Session.

Code example:

function team_add( $name, $team_website = '', $team_logo_url = '' )
{
    global $wpdb;

    $return = $wpdb->insert( DB_TEAM,
        array(
            'name'          => $name,
            'team_website'  => $team_website,
            'team_logo_url' => $team_logo_url
        ),
        array(
            '%s',
            '%s',
            '%s'
        )
    );

    if ( $return === false ) {
        throw new Exception( "Could not add team.." );
    }

    return $wpdb->insert_id;
}