The cudighi is decided by the spice blend in the sausage, not by the bun it sits in or the sauce that goes over it. Cudighi is a sweet-spiced Italian-style pork sausage, and the seasoning is what separates it from every other sausage sandwich: warm baking-spice notes, cinnamon, nutmeg, clove, and allspice run through ground pork, so the meat reads closer to a holiday roast than to a fennel-forward Italian link. Form that sausage into a patty, griddle it, lay it on a bun with melted mozzarella and a ladle of pizza sauce, and the spice is still the loudest thing in the sandwich. Strip the spice out and you have a generic sausage patty under cheese and tomato. Keep it and you have a sandwich that exists almost nowhere outside Michigan's Upper Peninsula.
It works as a sandwich because the patty form and the toppings are matched to the meat's behavior. Pressed flat on a griddle, the pork develops a seared crust that holds the spice against the fat instead of letting it cook out, and a patty rather than a link means even contact with the bun and a flat plane for the cheese to melt onto. Mozzarella is the right cheese here for the same reason it works on a pizza: it melts into long strands that bind the patty to the bun without adding a competing flavor, leaving the spice in front. The pizza sauce is the acidic counter, a thin tomato layer that cuts the sweetness of the spiced fat the way marinara cuts a sausage on a slice. The bun is a plain soft roll on purpose, sturdy enough to hold a hot, saucy patty through a lunch rush at a UP bar or drive-in but neutral enough to stay out of the sausage's way. Onions and peppers go on by request as the standard add.
The variations are small and stay inside the UP. A plain cudighi skips the sauce and cheese and treats the patty like a bratwurst on a bun. The pizza-style build, sauce and mozzarella, is the codified default most places mean by the name. Each of those deserves its own article rather than being crowded in here.