I think that today knowing one style is not enough. Here is a simple reason:
Organs you get to play might be from different period, different national school, and different type.
Imagine you only play Franck's music and you are invited to try out a small positive Baroque one manual organ. You would feel quite awkward not being able to play any chorale prelude by Pachelbel, a choral partita by Bohm or other pieces that works well for this type of instrument.
Or you get to play an English Romantic organ but all you practice are manualiter pieces by Scheidt. No pieces from his Tabulatura Nova would work for this type of instrument.
Or perhaps you are trying out a North German Baroque organ with a short octave and split subsemitones. Of course, you would need pieces by Scheidemann, Buxtehude, Tunder, Weckmann etc.
What could you play on an Italian or Spanish Baroque organ? Of course, Frescobaldi, Diruta, Banchieri, Gabrieli or Cabezon, Arauxo, Coelho and the like.
You might say that Bach's music would work on the organ of any style? While this is true for a lot of modern instruments, Bach's compositions wouldn't sound convincing in a lot o village churches which have 19th century organs.
You might also say that there is no chance you get to play such various historical organs? I wouldn't be so sure. In today's world, when people like to travel a lot for their vacations (and when connecting with local organists is just a click away), it's actually quite likely.
If you are travelling to another country and you don't have your music scores with you, the best way to try out a new instrument is simply to improvise. This is true not only if you go abroad but also if you are visiting another church in your town which houses an unfamiliar organ. Of course, the ability to improvise in various musical styles (with or without pedals) is still very important in such case.
You could also ask a local organist demonstrate the organ for you but this wouldn't be nearly as much fun as trying this organ yourself. This would probably feel like being on the beach on the sunny summer day and watching other people swim while you would be just standing there and thinking that too bad you left your swimwear at home (assuming you like swimming).