~다고 생각하다 is a combination of ~다고 (which makes the sentence a quotation), and 생각하다 (verb) = to think (of something). Appropriately, it is used to express your opinion of something, which is expressed as "I think..." in English.
~는 것 같다 has a nuance of uncertainty, so it can also be expressed as "I think..." in English, but the nuance is different as you are unsure. You can think of it as "probably".
같다 is not a verb, it is an adjective (to be the same). If you want to express "to be exactly the same" -> 똑같다. In English, we can express something that is the same as another thing as "like".
This apple is like an orange // This apple is the same as an orange.
A~(으)ㄴ B과/와/(이)랑 같다 = A is like B // A is the same as B.
You can use 같다 to express it smells like ~냄새 같다, it tastes like ~맛 같다, it looks like (you won the game) ~같이 보이다.