Thanks for the feedback. I did some more digging last night and came the same conclusion. These were probably local social clubs that may have had some support (official or unofficial) from the railroads. I also found on EBay (when all else fails, search there) where someone is selling a page dated late 1800s that showed a Railroad Men's Club that was built by the Vanderbilts for their employees.
As a former stock broker, I found it a bit odd that a non-profit would issue an actual stock certificate. Looking at it, this is not something someone decided to just print up so you can show you're a member. It has all the legal jargon and seals, and appear that everything was registered with the state under existing incorporation laws of the day. Like you, I have never seen a private club do this. Normally, you pay an initiation fee, then on-going dues. At the same time he joined this club, my grandfather was the secretary for the German-American Club in Jacksonville, and documents we have relating to that indicate just initiation fees and dues. No mention of "stock". I'm wondering if some lawyer talked the Railroad Club into doing it this way. A question we may never know the answer to.
This has given me the idea of possibly creating stock certificates for the operating group I'm in. Might make a fun Christmas present.
This is an interesting piece of railroad history, and hopefully one not totally lost to time.
As an aside, my grandfather grew up on a farm and believed only in investing in real estate. I remember him saying many times that if you can't go and physically see what you own, and run your fingers through the dirt, then don't buy it, and through out his life, he always owned multiple rental properties. Given that, this is probably the only "stock" certificate he ever put his name one (my grandmother, on the other hand, played the market and did quite well).
Modelling the Seaboard Airline in Central Florida