First: Yes, it is very likely. In the end, the MAC address is the only "reliable" authentication method available in the world of networks. However, it is also very likely that they use some kind of cookie that stores authentication information.
Second: a university server home server, huh? As I understand it, this server can never be accessed from the outside, no matter what you try, for a simple reason: clients from the outside have already turned away at the gates of the university. There is no way to accommodation. You need to ask the university network administrator to forward the port of your home server to the computer on which the home server is running. On the other hand, you do not need to authenticate yourself, as this is only applicable if your application establishes a connection. The server is waiting for incoming connections, so all this is good.
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