You don't need to register for a new K-ETA authorization every time you travel to South Korea.

It is not necessary because, once K-ETA is approved, it can be used for up to 2 years from the date of approval. However, if the expiration date of the passport used to apply for K-ETA has less than 2 years remaining, K-ETA will only be effective until the expiration date of the passport.

Check ETA requirements