In the TV shows or something, I've come across that kind of sentences. And make me wonder. why won't say just 'I pat his back'. it's shorter than 'pat him on the back'. and what situation would people say with the latter sentence 'pat him on the back' and could you give me more examples related to mine? (I hit him in the eyes.. or I hit his eyes.)
Looking at your examples, I feel that you are talking about some actions to be done on others.
Pat him on the back will be used with the preposition 'on' for a solid reason: it is an idiom and we use idioms as they are.
When you are performing some action on someone's organ, it has to be specified. Patting happens on someone's shoulder; hitting happens in someone's eyes considering that eyes were open then.
The safest way is to refer some dictionary and check whether it's an idiom/phrase. In other cases, I feel that the preposition is to be mentioned. Another example is kick on his ____ over kick his ____.
Praise given to a person for an achievement or a job well done.
Pat him on the back is would be better for this.
The coach patted each player on the back after the game.
The teacher patted all the students on the back for their good work.
Let's give them a pat on the back for doing a good job.
There's no real "why" in all of this. It's just how English is. And sometimes it can go either way. I can say "I punched him in the arm" or "I punched his arm." Other expressions are less lenient.
However, there is a peculiarity in the language around this. If you "pat him on the back," then you patted him. The body part is secondary information. If you "pat his back," then a morbid-minded person might wonder how his back had become detached from the rest of his body.
This is especially true with "head" and "eyes." If you "hit his head," there's a real possibility that he'd been decapitated, and you are just taking out your extra aggression on his now-lifeless head.
But ultimately, it's just the language. In English, I would say "I wash my hands." In French, I would say "je me lave les mains," which literally means "I wash myself the hands." So this linguistic thing with people and their body parts isn't unique to English. The linguists among us may even have a name for it.