Desdenova

Alright, I believe I've figured it out. It seems to depend on the resolution of the screen, which is set in the settings menu accessed from the main menu of the game.
When the resolution is set to 1280x720, like so:

then I can only see 2 Crew Portraits, like so:

However....when I increase my resolution to 1600x900, like so:

then I can add up to 4 Crew Portraits before the game prevents me from adding more! Like this:

And when I max it out to 1920x1080...

then I can see up to 5 Crew Portraits

 
So....yes, @jimmymcgoochie was right when they said to increase the screen resolution, I just glossed over that bit and only read the part about changing the UI scale. Guess that's a lesson in reading comprehension for me