I’m framing up a non-load-bearing wall that has two closet doors on it, close together (the king studs for the two openings are 13" apart). Unfortunately, they are aligned such that one of the 16" OC common studs would fall within that gap. In fact, it would only be 1/2" away from the left king stud, so not even anywhere near the middle of the space.) Is it still normal/expected to include that stud, or can I omit it?
(BTW: both doors will go to the same closet so there’s no perpendicular wall teeing into this one, in case that matters.)
For a non load bearing wall, the biggest consideration is going to be hanging drywall which is why you’ll see either 16" or 24" OC (to align with a 48" wide sheet). Since the wall isn’t meant to hold any weight above it, there’s a little more flexibility in where you place your studs, so you’re probably fine without that extra stud.
That being said, check your local building code, there might be a regulation for that.
Just to throw a wrench into this post: 19.2" is also valid spacing, as indicated by the mark on your (newer) tape measure.
24" spacing will likely require 5/8" drywall to prevent flex.