Why is a user allowed to have different names on different sites? It causes confusion and all.
It is allowed because each site has a different account. It is up to the users, to decide if they want to use the same username in all the sites, use a different name in a specific site, or use a different name for each site.
I personally use a single name because it's easier to synchronize the account information in all the sites. If I used a different name, I should edit the profile singularly, or remember I want to use a specific username in a site.
That said, there is nothing that forces users to use the same username in all the accounts.
As said by Oded, there should not be any confusion about knowing the username I use in another site, as my profile shows the list of the accounts I have in other sites.