Do I have to register for a new K-ETA authorization every time I travel to South Korea?

Not necessary because once K-ETA is approved, it can be used for up to 2 years from the date of approval. But in the case that the expiration date of the passport used to apply for K-ETA is less than 2 years remaining, K-ETA will be effective until the expiration date of the passport only.

Check ETA requirements