I owe you big time thank you 🙏🏻

사랑한다고 말해 주세요 is like "Tell me (that) you love me". It not quoting what someone said, but it would still be phrased with quoting forms in Korean. In Korean, the quote forms -다고 and -라고 are very broadly used. Anything that might be said or thought can be phrased as a quote, so pretty much any phrase that includes a noun-clause making "that" in English can be formulated with -다고 or -라고. It doesn't need to be something actually said. E.g. 그녀는 더 이상 견딜 수 없다고 생각했다 = She though she couldn't endure it anymore. 석유가 한 방울도 나지 않는다고 하는 나라 = a country which is said to produce not a drop of oil. 그가 위증을 했다는 사실 = the fact that he made a false witness statement. 몰랐다는 것은 변명이 안 돼 = That you didn't know won't excuse you.

Normal verbs and adjectives go with -다고. e.g. 친구는 떠난다고 말했다. 동생은 배가 아프다고 말했다. 이다 (is (something)) and -가 아니다 (is not (something)) go with -라고. 이다 has a function of coupling two things ("A is B") and 아니다 negating that ("A is not B"), so these two always come after a noun or noun phrase. e.g. 인간은 척추동물이다 (Human beings are vertebrates) => 인간은 척추동물이라고 한다, 고래는 물고기가 아니다 (Whales are not fish) => 고래는 물고기가 아니라고 한다.