The tokusatsu tradition of having characters be giant has largely to do with the tribute to kaiju cinema which preceded the likes of Sentai. Godzilla being the obvious touchstone, the largest practical reason of having Godzilla be gigantic - excluding the menace and the metaphors - is that the effects could look more impressive when portrayed on a massive scale. Similarly, as a departure from prior movies that used puppets and animatronics in the same era, the slow speed of Godzilla as an actor in a suit could still look menacing on a massive scale.
The departure can easily be seen from Godzilla's current menace, King Kong. While many clever tricks are used in both movies, King Kong achieves its massive presence through stop motion imposed on the film or occasionally giant still models. Godzilla, by contrast, is much more interactive with the events on screen and characters regularly interact with the consequences of its destruction. Moving this into broader tokusatsu filmmaking, while Sentai heroes could show off their effects more grandly at a 'normal' scale, there was a natural desire to showcase their influences by having them battle giant monsters (which also made it so they didn't always have to shoot on location).
The broader question here might be then why there is a mechanism for Sentai characters to grow large, or summon giant creatures to help them a la Power Rangers. Does this have its roots in Japanese folklore? It does not appear that there are any specific examples of this. I fully admit to not being an expert on Japanese mythology, but while there are giant creatures there does not appear to be spiritual mechanisms for summoning or controlling them.
In general, yokai (Japanese spirits) operated on their own principles. People in stories beseech to them for assistance, but they are not controllable by mortal means. Shikigami appear to be an exception which draws on a specific magical art, but these are more like curses than monsters in the physical world. The 'blue morality' of yokai do not lend them to being tools of humans, so if they were to become giant to fight other giant creatures that would be for their own reasons.
Now, onto Mario. What should be noted here is that Mario is not growing 'giant' when he eats a mushroom. That's actually his normal size, the only one that was intended to be in the original game at first. Through playtesting, the gang at Nintendo realized that by starting off as 'small Mario' and becoming 'big Mario', the image had more of an impact. So with that perspective it may be a bit more obvious that its not attempting to invoke the same feelings as a kaiju battle.
Shigeru Miyamoto (creator of the Mario character, designer of Super Mario Bros) has stated that the mushroom motif in Mario was similar to Alice in Wonderland, though I don't think he was ever clear on whether the Disney animated film or the book influenced him more. The opening scene in Wonderland with the "Eat Me" and "Drink Me" items shows that Alice is capable of different things at these different sizes, and likewise Mario has to tackle problems in different ways when he's small or big (I.E, when small you can fit through one block gaps but can't smash bricks).
Obviously creators do not always state every influence they have when creating something, but I personally see no reason why tokusatsu may have been an key reference to the original Super Mario Bros. The scale is not meant to be giant even though Bowser is bigger than the player. It's serving a gameplay function far more than attempting to evoke an epic scale and awe or danger to others. That's my opinion at least!
Now, later Mario games probably have been inspired by tokusatsu. Paper Mario: Color Splash has a Sentai-style Toad gang and Super Mario 3D World: World Bowser's Fury has Mario growing to an epic size for a kaiju fight against Bowser. These influences have trickled down to younger designers at Nintendo, but the origin of the growth mechanic does not appear to have been Japan's giant heroes or monsters.
Sources:
Japan's favorite mon-star : the unauthorized biography of "The Big G" by Steve Ryfle. Primarily a book about Godzilla but also the directors influenced by it at the start of the tokusatsu craze in Japan. It's well sourced which is unusual for a book of this type.
The Book of Yokai: Mysterious Creatures of Japanese Folklore by Michael Dylan Foster. An overview of the creatures in Japanese folklore and how they generally fit into these stories.
Iwata Asks: New Super Mario Bros: Volume 1. In this interview Miyamoto makes the connection between the Mario mushroom and Alice in Wonderland. He attempts to say it's not directly inspired, but I think this is playing more on the misconception of mushrooms as purely psychedelic substances. As noted in this blog post, Miyamoto had unambiguously stated earlier that Alice in Wonderland was a talking point at the inception of Super Mario Bros.