Double window has play when pushed on
It's hard to tell exactly what the problem is, but it certainly doesn't sound right. It may be that you have a mulled window (here's a video about mulled windows), and perhaps the flanges were not nailed in or the nails missed the (rough) sill or header.

I think that the place to start would be to remove the siding under (and around) the window to get a better look at the problem. Here's a quick video about working with vinyl siding.