So how do you know who owns a hotel? Would it be an individual or a group?
What does it matter? It's usually an investment group you've never heard of. The owners would either contract with the hotel company or an outside management company to run the hotel. As a guest you shouldn't be able to tell the difference.
And do they rent or lease the name Hyatt or Marriott, etc?
Do you understand what a franchise is? That's what most hotels are like. Just like McDonalds and Burger King don't own most of their restaurants, most hotel chains don't own most of the hotels with their name on them.
But the owners still have to adhere to certain rules and obligations to use the name Hyatt or Marriott or other hotels.
Yes, there are brand standards that the operators of the hotels must adhere to.
What does the hotel group get for partnering with the true owners?
Money.....they get a fee for the use of the name and what comes with it. (Advertising, reservations systems, etc.) They also usually get a percentage of sales.
When you book a room at a Hyatt or Marriott, how do you know that you are truly staying at a Hyatt or a Marriott?
Because the name is on the building.