Is it possible to cut a larger opening in this window / serving hatch of this pool house?
Yes, you can but you will need to properly support the structure above the new opening.

As you can see, there is a header in place now fashioned of two 2x4s with blocking.  This is not a particularly strong support but I doubt there is much to support beyond the roof itself.

Doing what you are proposing with the lines on the photo, however, leaves no room for anything but that one 2x4 which is probably not enough support.

You could probably get at least as much support as you have now with a couple of 2x4s and a plywood spacer between then.  You'd lose a bit of opening height from what you've drawn but it would be close.