You are correct in that you are not a FF on any air line just for having the Sapphire. Depending on what airline you are flying, you may be able to convert your Ultimate Reward points (earned with you card) to that airline's mileage program.

There are sometimes enrollment bonuses for signing up for FF programs, but I haven't seen a good one in a while. When you get to the airport, you can have them add your FF# to your itinerary just by asking.