I know bitcoin core is the most popular/suggested software to run a full node, but are there other options? I've heard of others like Bitcoin Unlimited. My question is, would those nodes running alternative software still have to have that software adhere to the protocol rules of bitcoin in order to be a full node on the network?
Each coin has its own Node and you can see blockchain as the soul... all of them have it's own blockchain and each chain have its own rules. And about the bitcoin network, there is no such thing, Bitcoin runs on the Internet on his own port, nodes are connected between them, but isn't a private network. The question is tricky, lol

To be clear, you can have multiple coin nodes on the same machine, for example, Litecoin, Doge And BCH.