Is there any difference in meaning between "to make up with someone" and " to make up to someone"? I've also heard " I'll make it up to him." in an American TV show. What role does that "it" play in that sentence? Does it mean the same as " I'll make up with(or to) him" ?
"Make up to someone" is incorrect. The other two are similar, but differ slightly.
"Make up with someone" is about resolution of a mutual disagreement or conflict. Perhaps both parties in a dispute, argument, or fight continue to harbor emotional negativity toward one another; one or both later decide to resolve the lingering issues, and one approaches the other for resolution. One person alone cannot fix a two-party issue, nor can one person alleviate both parties' negative feelings. As such, the "making up" (resolution) is a group activity, something you do together, hence the use of "with".
"Make it up to someone" is about a single person being wronged to some degree, often times to a second person's benefit. For example,
Hey man, can you cover my shift at work tomorrow? I have a friend in town and I'd like to meet up with him for a bit. I promise I'll make it up to you!
This request would inconvenience the person being asked, because they lose a day off should they agree to take the shift at work. But the requester is promising to "make it up to them" at a point in the future, perhaps by taking one of their shifts or with gifts or other favors. The requester exclusively benefited and the requestee was inconvenienced. Therefore the "making up" is not a group activity and thus we use "to" and not "with".
These are two different idiomatic expressions.
To "make up (with)" someone is to reconcile after a disagreement:
The couple was fighting all night but they made up before morning.
The estranged son made up with his father after his mother's death.
To "make it up to (someone)" means to offer redress for a wrong you have committed against them, or a favor they did for you.
Tom was late for his date, so he offered to make it up to her by taking her to a really nice restaurant the following week.
If you can babysit my son this evening, I promise to make it up to you sometime.
To "make up with someone" generally refers to an act of mutual forgiveness or sorting out of differences, like when a couple has a fight but they talk it through and as a result get back together.
To "make it up to someone" refers to an act of kindness or a favor done for someone because you have wronged them in the past. For example, if someone missed a friend's birthday they would "make it up to them" later by buying them lunch. "It" in that context refers to the bad deed that needs to be amended.
Actually, "make up to someone" is an informal expression used to describe someone who acts friendly or helpful in order to get some personal benefit. It is used to express disapproval.
For example, one might refer to a student who makes up to his or her professor to get better grades, or to an employee who makes it up to the boss to get on a fast line for promotion. (see: https://www.merriam-webster.com/dictionary/make%20up%20to%20(someone) OR https://www.macmillandictionary.com/dictionary/british/make-up-to)
To "make up with" generally means to reconcile one's differences with someone who one has fought with or had some sort of disagreement with. For example, after fighting for years, she made up with her mother and they now have a much healthier relationship.
However, I have also heard it used idiomatically for making plans, as in "I made up with Bill to see a movie tonight."
"Make it up to someone" means to do something nice to someone because they're upset (or you've upset them), or sometimes just because you feel that you've done something wrong as in: "I came late to Kate's party, so I made it up to Kate by staying after the party was over to help her clean up."