The reason that there is no widescreen / 1080p resolution available is because there aren't enough vertical pixels.

The game is looking for 1200 vertical pixels (1600 x 1200 resolution) for a widescreen display, but there are only 1080 available, so the game defaults to the 800x600 narrowscreen 4:3 resolution. Notice that the 1600x1200 option is greyed out in settings.

Source: Was staying away from home for the last month and couldn't bring my desktop, so I took my laptop and my desktop monitor, which is a 1440 display, and set that monitor as the only display. I bought Bridge Commander yesterday, installed it and it allowed me to select 1600x1200 resolution, widescreen and everything, no problems. Game ran fine, and I did a few missions and it was just as fun as I remember years ago. I got back home today, unplugged the desktop monitor and launched Bridge Commander, which refused to launch. I renamed the config file and the game launched to the default 4:3 narrowscreen 800x600 resolution, and I can no longer select 1600x1200. Plugged the external monitor back in, and widescreen is back.

So I guess the only solutions right now are getting a 1440 or higher monitor, or getting an old 1600x1200 CRT monitor.

Before buying a new monitor, you could check the graphics driver utility, like the AMD, Nvidia or Intel software.
I'm on AMD and there is an option in these settings (AMD calls it "virtual super resolution") which allows applications to render larger than the displays native resolutions and then be downscaled by the display driver.

Theres also a Remaster, that includes graphic options for widescreen:
www.nexusmods.com/startrekbridgecommmander/mods/4012?tab=description

Or a mod to just add 1080p resolution:
www.gamefront.com/games/bridge-commander/file/bridge-commander-resolution-mod

You are aware that this game is around 20 years old and not all the modern resolutions
were known at this time, do you?
Or better said the APIs did not know about these resolutions.
And yes, maybe they could change it.
But depending on the used APIs, just swapping one could break something big
in the game and more work would be needed to get this up and running.

You need to depend on mods, like the mentioned one, for such old games.

When using the Kobyajashi Maru mod (there are 2 versions) there is a button to use other resolutions, like 1080p. (Graphics, see bellow: alternate resolutions )

When i clicked on a resolution that was higher then my screen could handle, the game constantly crashed at start. Not only was it needed to re-install bridge commander but for some reason also gog galaxy and deleting the old gog galaxy folder manualy.

I tried BC on 2 different PCs. On the one pc it never crashes, on the other it still crashes regularly. Currently trying to figure out how to make it stable on the 2nd one.

see
https://www.gog.com/forum/star_trek_bridge_commander/info_bug_fixes_mods_installation_higher_resolutions