The company was named after the U+1F917 🤗 HUGGING FACE emoji.
HF is more of a platform for publishing this sort of thing, as well as the neural networks themselves and a specialized cloud service to train and deploy them, I think. They are not primarily a tool vendor, and they were around well before the LLM hype cycle.
It's complicated.
It's basically a forum created to venerate the works and ideas of that guy who in the first wave of LLM hype had an editorial published in TIME where he called for a worldwide moratorium on AI research and GPU sales to be enforced with unilateral airstrikes, and whose core audience got there by being groomed by one the most obnoxious Harry Potter fanfictions ever written, by said guy.
Their function these days tends to be to provide an ideological backbone of bad scifi justifications to deregulation and the billionaire takeover of the state, which among other things has made them hugely influential in the AI space.
They are also communicating vessels with Effective Altruism.
If this piques your interest check the links on the sidecard.