Hi!
Usually a partner university is a university that they have co-operated with for a period of time, and that they have developed a strong relation with. They work together on a variety of topics, and they often do student exchanges between their universities regularly. It is therefore most convenient to do an exchange with these universities, because your home university will be able to answer all your questions about them, provide you with whatever resources you may need, and explain the process exactly as it is. These partner universities know each others procedures, offers, what to expect, and more.
However, it is not impossible to do an exchange elsewhere. It only means that you will have to do a lot of the work on your own. For example, if you were to do an exchange with your unis partner university they will already have all the information you need about accommodation, classes, and other things. But if you decide to do an exchange elsewhere you will likely have to research a lot of this information yourself, unless you do it through a different agency. You will therefore have to serve as a communicator between your university and your desired institution, rather than it being the other way around.
My former university in the US had only partnered with a uni in Stockholm. One of my friends, however, wanted to go to France, and there was no issue with him doing so. He just had to do a bit more research and was a bit more on his own than the students who went the traditional route.
I would contact the international office/study abroad office at your university, and talk to them about it
They will definitely have information about how to proceed about studying elsewhere, and all the steps you need to take. You are definitely not the first one to ask them about that, so they will more than likely have all the information you need. It shouldnt be an issue, unless the study abroad program is very country specific. It really does differ from university to university though, so contact them and see what their policies are. Good luck!