The Sorting Hat had been determining the fates of countless generations of children in the magical world. As such, it was unabashed about ordering them about.
It represented the remnant of the Founders, and it was their school.
Putting tuition in the hands of the Ministry of Magic had been a mistake, in the opinion of the Hat. Its leading source of income was bribery, and those bribes were to cover for predation on the populace at large. Meanwhile, the farms and lands that had been the source of income for Hogwarts had been sold off, and the money pocketed by the same families, in many cases, that were currently bribing the Ministry. Nor had the population of Hogwarts expanded greatly, if truth be told. To top it off, the Hat had a headmaster to deal with who didn't care a fig for the opinions of the Founders, and was using the school as a vehicle for his own plans - making him in essence as corrupt as the leading families that controlled magic in "Wizarding Britain."
Both Hat and headmaster would be seen by an objective observer outside their conflict as being stuck in the era in which they were in their prime. The distance in social and magical and technological measures between the time of the Founders and the Victorian era of Disraeli and whale oil that Dumbledore had been born into and between his time and the current day were roughly comparable. That said, the Hat represented a more pragmatic and less self-involved frame of mind. Or rather, minds.
Being merely a Hat, it couldn't run about (no legs) waving a wand and impressing the populace. Furthermore, in comparison to the headmaster and the toady court that was the ministry, it was fairly shabby, certainly compared to the expensive and gaudy robes worn by Albus Dumbledore. In short, the Hat had started to show its age. Not that the headmaster had not, but the Hat needed every scruple of advantage it could dredge up.
The members of House Granger, of course, had no idea of all this.
Thus it was that when the Hat requested two articles of clothing from them and proposed to teach them both the art of tailoring, or as the Hat put it to Hermione, being a seamstress, they were dumbfounded.
Oh, and haberdashery, it added.
When Hermione had confirmed that all her clothes had been, naturally, purchased by her parents, the Hat requested she surrender whatever was her absolute favourite garment. In Harry's case, it demanded whatever his aunt had given him that was least objectionable, since the Hat was a bit mad, but not unreasonable.
Unlike the portraits, the hat wasn't obligated to tell the headmaster anything much, and it was sticking to that.
In the end, Harry had had to sacrifice his school robes, and now had only one set. This raised the issue of how the laundry worked, as Harry would have to use it much more often. The hat taught them how to make emergency robes with Transfiguration and Charms out of some of Dudley's rejects, robes that would last at least a day. But before that, the issue of house elves doing the castle chores arose. The hat gave Hermione's quibbling short shrift. The Hat knew these elves, Hermione Granger did not. The elves were happy at Hogwarts, and the pay they received in magic was priceless. To impose the silly, Muggle-created standard of currency on them was, the hat told her, cultural imperialism of the lowest order. By this point, the various shocks of the day had worn both children down, so Hermione could quibble no more.
When they'd gotten to the blueing in their Renewal of the Hat, which it told them was a genuine ceremony, if uncharacteristically informal, the hat told Hermione she had the Potioneer's Spark. "It's as clear as day," it added. Then it muttered, a bit indistinctly, "Strange I didn't detect it when you were sorted. Ah, well."
Thanks to their above-average magical gifts, their sincerity, and the optimism of the Hat, at the end of their work, it looked brand new.
"We now need to give you a very quick" - it noticed their yawns - "survey of what being in a House means. It's mostly the same whether you mean a Wizarding British House in or out of the Wizengamot, or a House in Hogwarts. All four founders have Houses in the Wizengamot also, after all."
"So far, you have the bare minimum for a House, an affinity, the fact that you are of opposite sexes, so the House can, in theory, reproduce itself, the ability to do magic in tandem, and what I sense as family magic developing already."
The sleepy pre-teens looked cheerful at hearing that.
"HOWEVER," continued the Hat, "you should look for ceremonies to cement the hierarchy, strengthen your bonds, and so on. You can't disregard the Potter name, but you might get away with Potter Granger - or take a page from the Spanish and call yourself Potter y Granger. Magic just wants to see you making the effort,"
That was the final initial thought they both went to sleep on. With that in mind, it's only fair to say that Hermione couldn't really be blamed for what went on later. It was, you see, all the Hat's Idea to begin with.