If built to code it would have a pan - might be a copper pan (most likely), but could be lead sheet also. That age could also possibly be flexible plastic sheet like heavy visqueen or hypalon, and that is right around when the first plastic and fiberglass pans came out, so quite a few types it could be.
Could be leaking - could also be leaking drain pipe or fitting, leaking water pipe or shower valve, or the drain fittings in the shower might be corroding (likely copper if that age) and leaking.
Could also be water coming through the surround (plastic, tile, formica, whatever) and running down into wall, then coming out on the kitchen side at the base of the wall.
Simple way to find out, assuming you cannot easily dig (or use fiber optic inspection scope/camera) to look into wall or ceiling to trace the leak - is to get a contractor (insulation, energy auditor commonly) with infrared camera, or rent one from Home Depot or tool rental place or some auto parts stores for about $50/half day, or use your smartphone or tablet with infrared scanning App (for many models, and many newer iPads and iPhones come with capability installed) and check for wall/ceiling wetness showing up.
Or have plumber do a shower pan test - they put rubber plug in pipe, fill shower floor with water, and let sit for 24 hours to see if it leaks.
Anotehr thing you can do - many kitchen cabinets have drop-in or only slightly nailed in "floors" - if you can remove the one where the water leak is appearing (assuming it is coming out from under cabinets) you might find the place it is coming from, assuming there are no kitchen pipes in the wall in that area that could be leaking.
