This isn't RAW, and I would discourage it.
A weapon with the Two-Handed property must be wielded with two hands to make an attack. A weapon with the Versatile property can be wielded in one or two hands with varying damage. It seems the designers intended these weapon properties to enhance game balance.
What your player is asking for is the benefits of a Two-Handed weapon (higher maximum damage and more consistent rolls due to the normalizing effect of the two dice) without the drawbacks of a Two-Handed weapon (occupying both hands at the same time), but the system already has Versatile weapons to fill that niche.
Although the player seems to be forfeiting the mechanical benefits and opting for flavor instead, and although I'm not suggesting the player will intentionally try to abuse this, I think it can lead to a slippery slope of unwanted interactions later on.
If the player eventually decides they would like to use a shield, you couldn't let them wield it in their empty hand without violating the Two-Handed property of the weapon, and that creates a weird dissonance between narrative and mechanics.
If the player wants to use an object on their turn in a fashion that requires an action, such as opening a heavy door or turning a crank, you may encounter issues with the action economy where the player can do more in a turn and be prepared for upcoming opportunity attacks than another player might.
If the player eventually wants to subclass into Eldritch Knight or otherwise cast spells, you may run into problems with somatic components. (This is debatable, depending on how you rule on somatic components when using two-handed weapons: more on that here.)
Suppose that the fighter loses an arm in the future during heroic combat while fighting with a Two-Handed sword wielded in the other arm. You would need to decide why the player can no longer benefit from the Two-Handed weapon property or else why they can bypass the restrictions of the Two-Handed weapon property when nobody else can.
If any other player wants to use "rule of cool" to wield a weapon with the Two-Handed property in one hand, it wouldn't be fair to deny them, so any of the above concerns could apply to multiple players.
If you establish the precedent that weapon properties and other game rules can be ignored for flavor as long as the results are mechanically identical, you run the risk of confusing and alienating other players and muddying the rules of the game.
Given the slippery slope problems, I don't see a good reason to support "rule of cool" in this instance. It is simpler to say "no, pick a different weapon" or "no, that character concept isn't viable" than to create the exception for them.
An alternative roleplaying suggestion.
However, I do have a suggestion for how to steer the player to an idea that may satisfy their character concept while offering better roleplaying opportunities and a better mechanical balance.
I think it is problematic for an able person to try to use a weapon designed to be wielded with two hands with only one hand. I think this would indicate that the Fighter has not trained well on their way to their class features, and I don't think it is realistic for a trained Fighter to intentionally fight as if they do not know how to use their weapon correctly or as if they are physically impaired. They would be utilizing a weapon in an unwieldy and awkward fashion for which it was not intended. Even historical real-world one-handed fighting styles with two-handed swords have been notably niche and controversial.
I would suggest instead that the player consider the character concept not of an able person making a choice to use a weapon in an awkward fashion but rather of a person with only one arm who has overcome significant challenges to become a skilled fighter despite the impediment. I would not scoff at allowing such a character to use a Two-Handed weapon with all of its mechanical benefits, since the player would be voluntarily opting into other mechanical setbacks to offset it, including the inability to also wield a shield or use somatic spell components effectively.
I do not foresee the player's existing character concept being a significant roleplaying hook because it doesn't truly impose any narrative challenges, but my alternative or a similar option establishes what I believe to be a much more compelling character background and offers more interesting roleplaying options.
My idea is only one possibility. In general, my suggestion is to work with the character to take the core of their idea (a unique fighting style) and flesh it out (a unique fighting style because of fill in the blank) into a form that can create roleplaying opportunities (a unique fighting style because of fill in the blank which challenges the character in some significant and recurring fashion). Challenging weaknesses tend to get more interest at the table than stylistic quirks.