The problem isn't the DM.
Inverse My Guy Syndrome is as much of a challenge to overcome as My Guy Syndrome. The tone of your question was (you to the other player) "your guy wouldn't do that" or "your guy couldn't do that."
Actually, yes they can and yes they could. It's the DM's place to set the boundaries.
Stats are descriptive, not prescriptive.
Unlike editions of the game where not having skill points might block you from attempting something, in this edition any player can have their character try anything, and the DM narrates the results. Dice are only rolled when necessary. (Basic Rules, p. 4).
The DM describes the environment.
The players describe what they want to do.
The DM narrates the results of the adventurers’ actions. Describing the results often leads to another decision point, which brings the flow of the game right back to step 1.
It is OK for a player to have skill
It is counterproductive to presume that as soon as the game starts the player somehow disappears and only the character is present - immersion into a given character in an RPG happens to the extent that any player desires it to. The level of immersion into "in-character", both preferred and experienced, varies with each player and with each table. The characters in an RPG only exist and have agency because a player is driving them for the game session(s).
- As but one example of how you can't avoid a bit of meta-play: the game feat Lucky, and the Wizard (Divination School) ability Portent, are Meta Game Constructs. They give die roll control to a player/character. The dice themselves are a meta construct outside of the characters who are inside-the-fictional-setting of the game.
A related meta problem is this2:
... mental stats {are} problematic in role-playing games because the whole point of a role-playing game is for the player to make choices and solve problems.
I have added an at-table example of player/character juxtaposition, as regards low versus high Charism - that troublesome stat - at the end of this answer.
Don't resent another player's ability - keep working on your ability
Mike Mornard once made the cogent observation - referring to the game Diplomacy's influence on early D&D - that player skill is a part of the game. It still is. The presence of min-max and optimizer play styles attests to that. So too does the learning curve of new players as they become experienced players. Player skill is a part of the game, even if it is seen by some as being part of the meta-game.
- When someone makes a spell choice on level up, what informs the reason for that choice: character growth or optimization? Are you going to complain about that choice to the DM, or tell the player that choosing that spell isn't what their character can, or would, do? I doubt it. And if you would, any player would be well within their rights to tell you to back off.
If the DM is happy to adjudicate the social encounter based on a clever bit of wordplay, good! That's fun and (1) it is good role play, (2) players play the game to have fun, and (3) in this edition you can try anything and see if you succeed.1
That third point is a fundamental premise of the 5e ability / skill check set up. Beyond that, stats don't rule all; see advantage and disadvantage rules as an example.
- Advantage beats stats
- In a case like this it can (depending on the circumstances) overwrite stats: if a given verbal exchange is particularly good or clever, it is a perfect place for the DM to apply advantage (+5 boost, roughly) to any attempt) due to a circumstance. Note: circumstances are not dictated by ability score and stats and this won't happen all of the time. To earn, for a given encounter, +5 is equivalent to the difference between an 18 CHA versus an 8 CHA (for that instance). Advantage can overcome that stat modifier if the circumstance set up by the player attracts that ruling from the DM. That's a part of this edition of the game. Disadvantage swings it the other way.
This answer is a challenge to the frame of your question, which has the embedded assumption that player skill should not to be taken into account by the DM. It is fine to do that at a given table if that's the table consensus. In the end the DM is the one who makes the ruling on the outcome of a player's decision, action, or verbal declaration/saying, as well as whether or not a situation accrues advantage / disadvantage.
Bottom line
The other player is not a problem player, unless they are being a spotlight hog and not giving you a chance to shine. That requires some player to player communication and teamwork to sort out. If that player generally hogs the spotlight, then as a fellow player remind them that you get a turn also.
Don't dump that on the DM; you discuss it with the player and you reach an accord with them. You both (your characters) are in this adventure together: work as a team.
As a practical step, the next time there is a social encounter, you volunteer to go first. If need be, tap that player on the arm and say "I've got this one" and do your best silver-tongued-devil thing. 😊
You are not the DM
Each of us who has DM'd, but is in the role of a player, needs to address the fact that our role is different as a player. We need to make peace with the instinct to try and back seat DM. (I still struggle with this sometimes). With DM experience, each of us knows how we would handle a given case as a DM.
Recommendation:
- You are a player: play your character as best you can.
- Enjoy the success of the other players, rather than seeing their good play through a competitive lens.
- Work with your fellow players to form a better team.
An at-table-example of Charisma, low and high ...
I was recently advised by my DM that my ranger, who has the lowest Charisma in the group, and whom I have deliberately kept out of a couple of social interactions in favor of higher Charisma players (once with disastrous results when the dice went very cold - heh-heh, the dice are fickle!) is seen as the group's leader. In character, that makes some sense since he's often made suggestions that the group adopts (we generally make decisions by consensus), we are in a forest and he's a ranger, and he has a relative in-game who is under a curse that he's trying to lift. He has a motive to try and get to an objective. If we were to confine ourselves to the highest Charisma character being "the leader" it would be socially dysfunctional (as players) to try and force that player into that role if he doesn't want it.
The players form a small social group and thus the party's reactions will frequently reflect the real world, rather than the in-fiction, social dynamic.
When I talk about making decisions by consensus I take it OOC as well as IC: I ask my fellow players for input on "what spell should I choose that will help the party the most?" when I go up in level. Ranger spell picks are limited. I want their input because I want the team to succeed both in-character - the ranger's relative, the ranger's goals - and out-of-character, as a player. Those two states are not contradictory.
1 The rules on Ability Checks state:
The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure. When the outcome is uncertain, the dice determine the results.
If the suggestion/persuasion seems to the DM to remove the risk of failure, then no die roll is necessary.
2 For further reading on how some ability scores are problematic, this rant (caution, language) is worth a look.