Why use use が instead of の here?

Hey, super beginner here.

I was watching this YouTube video that tests your ability to use particles among other things on an N5 level, and according to the video, you should say

わたしの好きな漢字

using the NO particle (which makes sense from an English perspective), but on the other hand you should say

かれが好きな馬

using the GA particle, which also makes sense, but is different from the first one.

So the question is, do you use the agent particle more when you talk about other people? Or is it because the object is animate (a horse) in the second example and not the first? Why use GA instead of NO and vice versa?

Thanks in advance guys.