I like him better/ more

Honestly, I don't see a difference. It requires the right context, of course, or we'll be asking "Better than/More than what/who?"

"Better" generally refers to quality whereas "more" refers to amount.
So, if I say "I like the red coat more than the blue one ", it will mean that my liking for the red coat is more than my liking for the blue one, which is understandable.
But, if I say "I like the red coat better than the blue one", it will mean that my liking for the red coat is better than my liking for the blue one.

Then you've answered your own question.

You asked if both sentences sound natural to native speakers, and if they have the same meaning. Yes, they do. I agree that if you analyze them literally, "I like the red coat more than the blue one" is more logical than "I like the red coat better...", but they are used interchangeably by native speakers. If you like, you can think of the "better" construction as an idiom.

Debpriya De1. I like him/it better.2. I like him/it more.Which of the above two sentences is more natural to native speakers ?Do they have the same meaning ?

I use both interchangeably. I don't see any difference. You might think of "better" as having an idiiomatic, rather than literal, use in this case.

CJ