WordPress can be accessed through two or more different URL’s, but it will only build it’s menus, permalinks, etc from a single URL. This means that if the site is set up as http://domain1.com and you try to access it from http://domain2.com every link on the site will show as http://domain1.com/page. That’s jsut how it works.
You say that you have another WordPress site that does this, so if that’s really working, look at what that site is doing and copy it – and then tell the rest of us so that we can do it in the future!
If the other site isn’t built in WordPress the two systems can’t really be compared like-to-like.
You are sort of correct, becuase it is mainly the URL setting that makes it work with only one URL. But you can only have one URL set at any time, so that’s where the restriction is. If you set up WordPress with two differnt URL’s you can still access the pages through both URL’s, but the links will only be to one isngle URL. You can’t pick and choose which one you want to use. You have to hard-code one. wordpress has always bene set up like this, and even though changing the system to allow multiple URL’s has been floated before, it’s not technically feasable to do it with the codebase the way it currently stands.
What is the actual reason that you need two different URL’s/IP addresses for this one site anyway?