"I like this bag better."
Translation:저는 이 가방을 더 좋아합니다.
This is true even if the 'more' is related to the noun eg.
물 더 주세요 please give me more water.
Are "저는 이 가방을 더 좋아합니다" and "저는 이 가방이 더 좋습니다" equivalent in meaning? (and is the second sentence's structure grammatically correct?)
I think not. You basically have two subjects in the second sentence ... unless "the bag likes you more". 좋다 here is transitive and needs an object but you have 가방 tagged as the subject and 저 placed in sentence order to suggest it is the subject.
I had the same questions as Mystic_Fire. But I think Mystic is correct here. The same question on italki: https://www.italki.com/question/144030?hl=ko
좋다 means "is good" and is connected to 가방이 in this case. I think i have seen somewhere saying that the second sentence traslates literally to: "For me, this bag is better" meaning -> I like this bag better.