Is it possible to play this game on 1920x1080?

No, at least not out of the box. It is an early Direct3D game from the old times when most games still ran in 320x200/240, and 640x480 was considered "high resolution" for games, so that is the highest the game supports.

The game options have an option for a 16:9 aspect ratio, but that just adds silly black borders on the top and bottom of the image, still showing the image on a 4:3 aspect ratio. So the game does not have true 16:9 widescreen support.

I am not sure if there is some user-made mod that increases the resolution and adds also widescreen support, but with these kinds of old games increasing resolution doesn't improve the graphics much because of the low polygon count and low resolution textures. So there are no extra details that would become visible with a higher resolution, like there is with modern games.

You may be able to use dgVoodoo to force the resolution up, as long as you're playing in Direct3D mode. Even if it works though (I would say there's a 40% chance of it working), I would stick to 1440x1080 or 1360x1024 as a resolution, as forcing a 1920x1080 resolution won't give you proper widescreen, it'll just stretch the image.