Is there any way that I can integrate the UserCake user management system with MediaWiki? I want to link the accounts in each system so users can log into both with the same username and password.
There is no existing system to do this. You can create an extension for MediaWiki to do this without too much trouble. Basing it off an existing extension is probably a good starting point. By looking at AuthJoomla, AuthBugzilla and AuthSymfony you should get a good idea of how MediaWiki's authentication extension API operates. It's really just a matter of creating the class, getting it to call the relevant UC functions and then loading it as an extension into MW.
Having said that, UserCake seems unmaintained. It may be worth taking this opportunity to migrate your users to another authentication system. If you are using credentials between several applications you might want to look at something like LDAP.