Until last year I was a licensed agent in Florida, before l moved here to North Dallas. I can say that what you have described is a lot more common there. In Florida and many other states, they are different types of listings, Exclusive Right to Sell, Exclusive Agency, and Open Listing.
Exclusive Right to Sell, is the most common, it give the agency the right to market, and represent the Seller during a transaction. As well as earn a commission if you, another agent, or anyone brings the buyer. The listing agent then agrees to Co-broke, or cooperate with the other agent, in other words split the commission.
Exclusive Agency primarily the same, but with the major difference being that you reserve the right to sell your home yourself and not pay the broker a commission. The broker only gets paid if your home is sold through them OR if you allowed them to enter it into the MLS and then the broker must Co-Broke with another licensed real estate agent. These are really not a safe type of listing for an agent, unless they really trust the seller. Also agents who take these are acting out of disparity to "Get the listing" and may work extra hard. But also you have the ones who just want the sign in the yard and do nothing to get it sold.
Open Listings are funny! It gives the agent/broker permission to show potential buyers your home, and the broker will only earn a commission by bringing in a client who buys the home. Since the open listing isn't "Exclusive" sellers can sign these listing agreements with as many brokers as they want. So picture this, 10 real estate signs from different companies all in the front yard. But only one agent would be able to agree to do the entry into MLS, and then that agent must agree to co-broke with the other brokerage.
It's amazing to me that a lot of licensed agents do not even understand that about the MLS.
Now with all that being said, the reason you may still see that listing on the site is a lot of those FSBO sites, are slow, incompetent, and are as bad as a lot of agents...they just want the numbers on their sites. Plus a LOT of FSBO websites, such as "For Sale Buy Owner
that is very popular in Florida are actually just a marketing company and NOT acting as Real Estate Brokerage.
Also, they can ACCEPT other offers, they just may not be able to EXECUTE it, hence why there is a special back-up addendum to go along with offers.
Good luck, and let me know if there is anything else I can help you with.