I'm a little bit confused here because my teacher said we need to use 옆 when we want to say something's on the left/right and that's how we've been doing it in class :S

I don't know why your teacher has said that but at least to my ears it is redundant at best and wrong at worst to say that. However, I am not well-versed in Korean grammar (never studied it, for obvious reasons) so I recommend you ask on other websites/Discord servers. All I can say is it is either 옷장이 문 옆에 있어요 or 옷장이 문 왼쪽에 있어요.

My best guess is that maybe it's different depending on which part of Korea you're from? I honestly have no idea :(

I asked this question on a Korean learning Discord and they all said it is redundant/wrong to say that and you might have misunderstood your teacher's explanation.

That's what I thought too but I was just rewatching the zoom class and she wrote 옆 after 왼쪽/오른쪽 several times during the 2h class... I will for sure ask her about this.