Probably just the plaster. The bottom coats of plaster aren't the nice pretty white finish you see at the surface.

As for moisture resistant, you've seen what happens when it gets wet.

They make a pretty cheap and easy paint on waterproofing membrane for showers though. You paint on a couple coats and then tile right over it.

Couple of questions, do you know what type of walls are in use in the rest of you house. In construction from the 30's they applied a cement type of material on typically on a metal mesh for bathroom locations. I can't say I've dealt with 40's bathrooms.

In my former mid 30's home the previous owners had removed old tiles and retiled over the base. For the most part it held good expect for an area where they had patched the opening for the recessed soap dish with a non water resistant material. I used a concrete patch to fill in the gap, and then used tile repair material available at most home stores. I had a tile guy come in for my 1/2 bath at that home to determine how to deal with some cracked and loose tiles and he mentioned that product is quite strong and it doubles as grout too. Just follow instructions.

I have plaster walls in the rest of the house. The material behind the tile doesn't look, or feel, like the plaster in the rest of the house (it seems stronger than the plaster). I've looked into an adhesive and grout in one, though the person at the hardware store said it was really only suited for small jobs (4-5 tiles).

So long as there are no cracks or loose areas of the mortar, you couldn't have a better subsurface to set new tile on. it's an old fashioned mortar bed, and I guarantee you, you'll pay hell getting it out, even with dynamite!