I feel like the terms of groups is very confused within the community and id like to clarify to what each is to my understanding, to do so I'll be talking about Guilds, Schools Business's and associations.
"Guilds in Chronicles of Elyria act as professional societies and unions specializing in a specific craft. Guilds fulfill a wide range of activities that may include facilitating contracts of trade, development of new technologies pertaining to the craft, and patents of new discoveries on a broader scale than can be achieved through individual efforts alone." - Wiki
In my opinion one should expect a guild to be, and more specifically a guilds needs to be something that allows the research and production mechanics to be relevant, also known as crafting skills and professions like blacksmithing and Leather-working.
Guilds will provide perks to production and research as well as allow new technology to be patented. That being said i think its practical to say only groups concentrating on crafts and professions capable of research and production advancements can be a "guild" since only they will make use of the mechanics.
A good example of a common mistake in CoE standards of groups is "assassins guilds". That doesn't make sense in this game, it would need to be an "assassins association" or other variation of groups.
"Schools in Chronicles of Elyria represent a collection or group of people who share common characteristics and goals of a certain philosophy, discipline, craft, economics, art, or culture." - Wiki
Much like a "guild" schools offer mechanical perks to things like research and learning. However one of the bigger differences is that "schools" will have a greater perk to research than "guilds" that, in return, receive a greater perk in production.
Another difference between a "guild" and a "school" is that any research done in a school is considered free knowledge, and will not be open to being patented for personal use by the institution or anyone for that matter.
It is important to note that, unlike a guild a school has a learning perk mechanic that makes it useful to a wider range of activities and professions that don't benefit from a production or research mechanic like Astronomy, Cartography, Music, Scholarship, Scribing, Art and Cooking and other things that may be "learned". Much of these are things under the bardic skill tree that is often seen as the tree a "scholar" would go into.
A business in Chronicles of Elyria leaves endless options and opportunity in the hands of the players, although there are no in-game mechanics for this type of group (aside from contracts) it will be a big part of the game. Similar to the real world, if you can create a demand for something, whether it be a product or a service, you can make a business of it. This ideology will be the same in CoE.
Some examples are Bank, Transportation service, restaurant (tavern), sellswords, Inn, mining, skinning, and more. The game leaves endless options in the hands of the players.
Associations are one of the variation of words that is used to describe a group that doesn't fall under any of the 3 groups, they are not a Guild, School or Business. An association would be a group that works for a certain cause or towards a certain goal.
Some good examples of associations are "Assassins Association" or "Human rights Association". Its a loose term of a group of individuals that organize themselves for a common cause or goal.
Finally, to answer one of the OP's questions, yes this does mean you can technically make your own religion, granted you may be limited; however, you can very well gather a following of people that worship and follow the guidance of your "made up" religion and act on the beliefs of that religion. It would be a group of people gathered for a common cause, belief or goal.