It is possible to have a site aliased as both:
shopping.mysite.com
mysite.com/shopping
I'm comming to the conclusion that the answer is no. If this is possible how would I setup the links. If it just use the link tool in the editor. The link will be created relative to where I am. For example, if I create a link to page1
shopping.mysite.com link is created as /page1
mysite.com/shopping link is created as /shopping/page1
And then the link will only work in the site it was created for. Other site is 404.