When I start the game it lets me select the screen resolutions of 640x480, 800x600, or 1024x768. When the game starts there should be black bars on the sides to maintain aspect ratio but I notice the graphics fit the entirety of my widescreen monitor.

Normally when a non-widescreen game stretches out like that it looks squashed or "stretchy" however with this game its hard to tell.

Does this game run in widescreen automatically? If not, is there a way to force it to maintain aspect ratio? All of my other non-widescreen games maintain their aspect ratio.

With the default install from GOG the game is definitely stretched.
They probably chose that default because, as you say, it's not terribly noticable.

If you want to change it you can try the dxcfg.exe in the install-folder.
Here is what worked for me:

Display mode: same as desktop
Aspect ratio correction: Enabled (enabled == 4:3, disabled == stretched)
Presentation: Windowed (yes, windowed. Don't ask me why. The game will not run in a window, but selecting Fullscreen made the main-menu non-functional for me)

Leave everything else as it is.

Oh my, this game runs perfect now. I did not know about that dxcfg.exe. Something I'll keep in mind for other games that might have it in the future. GOG community helpful as always. Thanks! :)