You can use the same application info for more than one site if one of the following applies:
– All the sites are subdirectories under the same domain name (foo.com/a, foo.com/b, etc).
– All the sites use subdomains and you specify the Application Domain in the Facebook app as the bare domain (entering “foo.com” as the App Domain will allow a.foo.com and b.foo.com to both work as well).
– Each separate domain name is entered into the Facebook Application as the App Domain (foo.com, bar.com, both entered into the application).
Basically, the Facebook Application must be able to recognize the domain of the current page that you’re viewing as being valid. Note that this applies to both the front end of the site as well as the wp-admin back-end, meaning that if you have a setup where they are on different domains, then the FB Application must know about them both in some fashion.
If the FB App doesn’t know about the domain, then it will fail to authenticate properly and stuff will seem to be randomly broken.