The sense you are thinking of, "to learn (for the first time) by reading", certainly exists; OED:
III. To learn about by reading.
b. intr. With of. To find mention, record, or discussion of in the course of reading; to learn about by reading; to read about.
I agree with you and the above commenters that this sense must be perfective, not progressive.
However, there is another sense in which you can say "read about", as noted in the comments, where "to read about X" means "to read (something) on the subject of X".
I don't think it would be considered a phrasal verb. Rather, we begin with a general intransitive "read":
— "What are you doing this evening?"
— "Oh, just reading."
This is then followed by the common preposition "about" as in "on the subject of".
— "You're reading, eh? What about?"
— "About Nelson Mandela. Did you know he didn't actually die in prison?"
(Notice that for a real phrasal verb like "pick up", you could not say "You're picking, eh? What up?")
Since this collocation is arguably not a fixed unit, it's somewhat hard to find direct reference to it in dictionaries, but we can spot it everywhere. This Collins entry could apply:
12. to gain knowledge by reading
He read about the war.
Similarly, this Free Dictionary definition is in the right direction:
read about someone or something
to read information concerning someone or something
The two examples given are of different senses, unfortunately; the first is your original, and the second is very likely this other sense:
Did you read about John in the newspaper? (i.e., did you find out what happened to him?)
I read about bonds, and learned a lot about finance. (i.e., I read on the subject of bonds)
We can also find a nice, clear reference to the extremely similar phrase "read up on/about" in the Cambridge dictionary, although they do call it a phrasal verb because they include "up":
read up (on/about) sth
to spend time reading in order to find out information about something
It's a good idea to read up on a company before going for an interview.
But as the users in this other ELL thread write, the two phrases work perfectly well without that "up" (at least they do in AmE):
I've read a lot about this topic lately
I've read a lot on this topic lately
The use of "a lot" there is an excellent clue that it can't be the "learn about for the first time" sense.
Here's the phrase in action in a Toronto Star headline:
Friendships like these … read about them in the newest books
Note that you don't read about "friendships like these" as if they were news items, but as themes of novels.
For me, this is satisfactory to show that this sense exists. Now, if the intransitive verb "read" can be progressive ("Don't bother me! I'm reading"), then this collocation can also be progressive.
— "What were you doing when I called you to dinner?"
— "I was reading about sharks in Nat Geo. It says they can change their sex at will! Is that true?!"
This Ngram shows that "reading about" exists and is at a sizable ratio compared to the standard form "read about". (Admittedly the gerundive form is included in there too, but I can't imagine that would affect the numbers too much.)
Finally, there's the conversation you cited:
Kathleen: I'm reading about proper etiquette while I'm there.
To "read about etiquette" in your original sense would suggest that the speaker never knew that there was such a thing as etiquette in Sauda Arabia. But that would be quite surprising. Instead, she's boning up on the subject to round out her knowledge and avoid making faux pas.
P.S. I will say, though, that the sentence is very slightly awkward because "while I'm there" is phrased as though it would modify "I'm reading about", but it evidently can't because she's reading it right now according to the conversation and she's going next week. Instead, it modifies "etiquette" in a somewhat colloquial way: it's elliptical for "I'm reading about the proper etiquette [to employ] while I'm there". In any case, the awkwardness has nothing to do with "reading about".