It was occasionally a thing. There are various examples of a reverse grip being used with a sword, from various places and times. They tend to be special-purpose, and the movie/game version doesn't appear to be based on the real versions.

First, note that reverse grip (i.e., icepick grip) is very common for daggers. For example, from Talhoffer: https://commons.wikimedia.org/wiki/File:De_Fechtbuch_Talhoffer_169.jpg

If you want to use a sword (probably a short sword) to stab in icepick grip like a dagger, it would work. Icepick grip was used with quite long daggers - in the Talhoffer example above, the dagger points reach past the elbow, and the daggers might have 16" blades. The Ammendola sarcophagus, c. AD160, shows a Roman example of a gladius being used in icepick grip: https://commons.wikimedia.org/wiki/File:Sarcophagus_with_battle_scene_DMA_T42492.jpg

Longer swords are less likely to be used in dagger-style. But longer swords are sometimes shown used with reverse grip:

• For parrying a spear in order to close in to dagger/half-swording range: https://commons.wikimedia.org/wiki/File:Ms.KK5013_01r.jpg from the Gladiatoria fightbook: http://wiktenauer.com/wiki/Gladiatoria%28MS_KK5013%29

• While wrestling in armour: http://wiktenauer.com/wiki/File:Ms.XIX.17-3_18r.png from Talhoffer: http://wiktenauer.com/wiki/Hans_Talhoffer/K%C3%B6nigsegg

• Cheng Zongyou's 單刀法選 Dan Dao Fa Xuan includes one technique where the sword (a long two-handed dao) is drawn in reverse-grip in order to parry a spear thrust, after which the wielder switches to a regular grip. See "translation preview #3" at https://www.chineselongsword.com/longsaber for this example.

• It also appears in the Korean military martial arts manual Muye Dobo Tongji in the long sword and admiral's sword forms. The position is called "Shikkum Sajukse (Drawing the Sword and Watching the Enemy Posture)" (spelling as per Kim translation), so it's intended for use as a defensive stance immediately after drawing the sword. Note that both swords for which the technique is used are quite long, so these are Korean analogs of the Chinese long dao method above. https://books.google.com/books?id=sIS_8u2x_ioC&pg=PA134&lpg=PA134

• Similar to the above, some Chinese sword forms (e.g., Chen taiji jian) begin with the sword in reverse grip, and the first few techniques are in reverse grip. These techniques are blocks, and represent either the just-drawn sword being used to block, or possibly the sword still in its scabbard being used to block.

• At least some iaido kata have the sword switched to reverse grip for re-sheathing.

The above are all techniques for blocking, wrestling, and resheathing. Cuts and thrusts with these longer swords all use conventional forward grip. For possible offensive use of reverse grip, we can look at butterfly swords ("baat jaam dao" in some styles), twin short dao. In some styles, these are used, part of the time, in reverse grip. The techniques in reverse grip are variously interpreted: as cuts, as blocks, as representing the weapon just drawn. At least some forms like this have the swords switched to forward grip for obvious cuts (so the reverse grip techniques are likely to be blocks followed by cuts). These forms do not present us with a clear-cut case of historical use of swords being used like the movie/game style of reverse-grip wielding.

EDIT: Korean example added.