Hi,
So from what that person says the problem is not with the customer details on the order.
It's with the cart content and the user information displayed on the website.
This looks like a caching issue. For example, if you configured a caching system to cache all the pages of your website, it's possible that different people see the same content as long as the URL of the page is the same. Also, individual modules can be cached and produce similar effects.
I would recommend to turn off the caching that you have on your website.