If I access chat from Meta Stack Exchange then tap on the "favorite" tab I just get a blank screen.
This is not the case for Chemistry Stack Exchange or a couple of other daughter sites.
As animuson commented, Meta Stack Exchange has its own separate chat server (chat.meta.stackexchange.com), with its own separate chat rooms and overall system from most of the Stack Exchange network sites (chat.stackexchange.com).
The reason why you don't see your favorite rooms from other sites there is because you only see favorite rooms from the same chat server. As most sites share the same chat server, you see favorite sites by accessing chat from a different site because they're also hosted on chat.stackexchange.com.
As an aside, if you tried accessing chat from Stack Overflow, it also wouldn't work, because Stack Overflow also has its own separate chat server, chat.stackoverflow.com.