That isn't a floor drain pipe. It is an electrical conduit. The drain pipe should be in the white box on the wall.

I am clearly not an expert, but thought this was the "water pipe" where the washer needs to drain IF there's a leak/flood. Based on your suggestion, what electrical parts would go inside this?

You need to have a qualified contractor take a look at that. There should be some caulking, flashing or other material to make certain water that does not flow into that pipe does not then go straight to the ceiling to the room below the laundry room if your washer does leak. You are missing the pan that the washer would sit in that should be part of that drain. Also, is there indeed a drain the box on the wall? I could not tell. And, lastly, is your new dryer gas or electric? If gas you are okay with the electrical connection.

It does appear to be for an overflow floor pan in which the machine sits. A washer’s drain hose needs a standpipe such that the end of the hose is above the machine’s highest fill level to insure siphoning doesn’t occur.

Look in the valve box and you'll see where the drain for your wash machine goes please. With the installation being on the 2nd floor, it makes more sense that is a drain for an overflow on the floor.