Are there any analog error correction codes that are additively and multiplicatively homomorphic?
Such codes are known as fully homomorphic encryption (FHE) because once you can add and multiply then you can calculate arbitrary function. FHE is a big challenge and it still does not have a simple solution. You can find even workshops devoted to FHE.
Usual FHE contains errors. The number of errors increases during computations and it necessary to remove these errors from time to time. In this sense FHE is an error correction code.