Here is one of the great things about being a DM. You can chose to be fair and make your own ruling on these types of things. You can spend your game trying to come up with excuses to cheat players.
The feature does what it says. Since the Giant Might ability clearly says "you become large, along with anything you are wearing." Anything you are wearing, this includes carrying, is now a size larger, and we now need to know if this change affects the weapon in some way, Thus, we now need to know what a large size, or over-sized weapon does for damage. The Dungeon Masters Guide has rules for oversized weapons on pages 277-278. It begins by talking about monsters who have oversized weapons and how to calculate for oversized weapons. This should definitely include your currently over-sized (large) weapons, making them one level of over-sized, which by RAW (Rules As Written) doubles their dice.
If your DM makes it all about "oversized weapons only apply to monsters", then do what a guy in the campaign I am playing in did, and chose a "monstrous" race like the Bugbear, Centaur, Deep Gnome, Duergar, Fairy, Firbolg, Githyanki, Githzerai, Goblin, Grung, Hobgoblin, Kobold, Lizardfolk, Merfolk, Minotaur, Naga, Orc, Satyr, Siren, Vampire or Yuan-Ti. All of these are races from the Monster Manual, so all of these would qualify, even by the DM’s twisted interpretation of RAW. In this example, the player literally has a Minotaur with a greataxe.
There is no way you can tell me that when a Large size Minotaur with an vicious, large-sized, two handed Greataxe charges to attack you, that it isn't going to cause massively more damage than usual. Any other answer than "yes" and I call, "BS" !
Finally, the best solution to this mess that gives you large-size weapon damage that is impossible to ignore. Take Smith Tools, or hire a Smith, and make a large weapon. You can also get large weapons for free off dead creatures of large size (bugbear, minotaur, Fiends, golems, etc.). Carry the weapon with you, pull it out, put it on the ground and lean it against your body/leg. When you grow, grab the large weapon that already has the increased damage and use it. Now there is no way, without being the world's greatest hypocrite and cheat, the DM will HAVE to give you the appropriate damage.
You can even then ask, after that combat, what is the difference between the large weapon you brought, and the large weapon you also hold, that grew with you. At this point it is fun to watch people dancing to justify their contrary logic/ruling.
And for real-world value, I know of at least 6 tables that rule it this way, so their are quite a few DMs out there who agree with this ruling, you just have to look for them.