I don't even know what these exhibits are called, but generally, it's a shop in a touristy area where you can try on all kinds of different costumes, and then have your photo taken with a kitschy background.
What are photo places like this called? Is there a place like this in San Francisco? (Doesn't have to be Old West themed).
These photo shops are generally termed Nostalgia Photography Studios, which sometimes include 'Costumes' in the name. There are literally hundreds and they exist wherever there are tourists, and fun daily activities for children. They go under a number of company names, such as 'Old Time Photo and Costume Co', 'Old Time Photos'.
Google will present for you those in your area with a simple query:
nostalgia costume photography CityName
You can hire photographers in the Bay Area to work at your event or party but there are no shops which offer this style of photography in the Bay Area. Santa Cruz Beach Boardwalk used to have one, but it is now closed (2014).
The closest I was able to find is Doc Wenzel's Old Time Portraits in Monterey. A regular sized photo is $16, and you are expected to purchase at least one for each person in the shoot. You get discounts if you buy extra. There were shops in Sacramento as well, but I don't know their names.