I am facing the same issue with the select box options not displaying when recording. Currently, I use the OBS Studio version 30.2.3.
This issue happens when choosing "Window Capture" for the capturing method. The select box options display well with the "Display Capture" method.
To better showcase this issue, I recorded how I captured the screen with both "Window Capture" and "Display Capture", along with the resulting video files. I attached the mp4 files, and the HTML file I used in the video:
Google Drive Link.
For a temporary solution, I change to "Display Capture" when I have something to record that needs to show select box options. However, at the moment I cannot use "Window Capture" when I'm just trying to capture a window only and I have to show the select boxes.