I freeze shredded cheese all the time and it freezes and thaws out just fine.
Sometimes the grocery store here will have a sale on some shredded cheese and I buy as much as I can and then freeze the excess and it lasts for a few years frozen.
I usually use the shredded cheese for baking things such as casseroles and melting it on nachos and anything else that I want to melt the shredded cheese on.
It's recommended to use the frozen shredded cheese within a few months for the best freshness but it's still very safe to cook with and eat after a few years as long as it's been frozen.