Yeah, you could say like that. It is correct but I would use "interests", "hobbies", "tastes",etc.. "People have different interests", sounds more natural with me.

I would use the word "interests". Usually when we used the word "likes" in English in that context of a sentence, we would say "people have different likes and dislikes." The word "likes" on its own does not end the sentence well. I hope this helps, it is the best way I can describe it.