Can I adjust the aspect ratio on a Nintendo Switch to display on a 4:3 monitor?

No, it doesn't seem so. It seems Nintendo imagined that people would only connect to 16:9 displays and did not add any options in the Switch display settings.

I have been having the same trouble for months. I have my switch hooked up to a 4:3 CRT TV via HDMI to RCA converter. It may be possible to find an adapter that can translate the image to show at a proper aspect.