I think it is a rulings not rules situation that is not defined RAW in 5e but with an implication and base of player autonomy over the character, but there is ambiguity about using skills on PCs.
5e PH Page 6 says "2. The players describe what they want to do. Sometimes one player speaks for the whole party, saying, "We'll take the east door," for example. Other times, different adventurers do different things: one adventurer might search a treasure chest while a second examines an esoteric symbol engraved on a wall and a third keeps watch for monsters. The players don't need to take turns, but the DM listens to every player and decides how to resolve those actions."
Page 185 says "Roleplaying is, literally, the act of playing out a role. In this case, it's you as a player determining how your character thinks, acts, and talks." It does not say except for magic and it does not mention skills one way or the other. Neither are addressed there.
It also says "The DM assumes the roles of any characters who are participating in the interaction that don't belong to another player at the table. Any such character is called a nonplayer character (NPC)." It then goes on to discuss PCs trying to influence NPCs.
on 186 it says "In addition to roleplaying, ability checks are key in determining the outcome of an interaction." This does not say whether it is restricted to an interaction with an NPC only and not with another PC.
It does not seem to address the use of social skills on a PC outside of not mentioning the situation at all.
Similarly in the skills section.
The section on contests under using ability checks on page 175 seems like it could cover player versus player actions.
"Sometimes one character's or monster's efforts are directly opposed to another's. This can occur when both of them are trying to do the same thing and only one can succeed, such as attempting to snatch up a magic ring that has fallen on the floor. This situation also applies when one of them is trying to prevent the other one from accomplishing a goal-for example, when a monster tries to force open a door that an adventurer is holding closed. In situations like these, the outcome is determined by a special form of ability check, called a contest.
Both participants in a contest make ability checks appropriate to their efforts. They apply all appropriate bonuses and penalties, but instead of comparing the total to a DC, they compare the totals of their two checks. The participant with the higher check total wins the contest. That character or monster either succeeds at the action or prevents the other one from succeeding."
Under page 178 for Charisma checks it says "A Charisma check might arise when you try to influence or entertain others, when you try to make an impression or tell a convincing lie, or when you are navigating a tricky social situation."
There is no explicit limitation on others. It does not say whether others means NPCs only or anyone other than you including PCs.
None of the charisma skills say explicitly they only apply to NPCs, they use language like "someone" or "audience".
The DMG gives guidelines a DM can use for NPC reactions and does not address PC to PC use of social skills or NPC use of social skills on a PC.
All of the social skill things are not defined as hard mechanics, they are usually phrased as a DM could call for a check, etc.
A DM/group would be completely justified under 5e RAW in saying a skill check does not tell a PC how to act, they determine their own thoughts and actions. There is also support for an interpretation that a skill check can affect PCs and there can be a table understanding to roleplay out check results the same as if magic induced an influence on the character.