When describing the people or things which appear in a picture, we use "in".
If you were to use the phrase "on the picture", it would most likely be to describe something that was not part of the picture originally, but is now on it.
"Your coffee cup is sitting on the picture!""I left out the crayons, and my toddler drew on the picture you left.
So sorry!"
ETA:
In North American English, the example you gave would be ungrammatical.
I see the book was published in England, though.
Perhaps it is considered an acceptable construction there.
ell.stackexchange.com/users/3175/michelle
I have never heard the phrase "on the picture" to mean "among the things|persons captured by the camera here".
For that meaning, it is always "in the picture".
Who is this person in the picture?
One can say " shown on the picture" to mean "which appears in this image".
Point to the strange growth of foliage, shown on this picture, which you believe to be the gangster's camouflaged hideout.
One can write an essay about a painting or photograph or movie, and say that one was writing an essay "on the picture", i.e. "about the picture".
ell.stackexchange.com/users/10264/Tᴚoɯɐuo
This is the same in British English as in North American English, so the example sentence is ungrammatical - the explanation given by michelle applies equally to both.
As a native speaker, I would assume that this is not strictly a grammatical error (meaning the author believed it was correct), but a typographical one (meaning the author did not notice their mistake).
Since you mention seeing it more than once I would suspect a copy/pasting error.
(It may seem odd for this to be missed by the editor, but these things do happen.)
ell.stackexchange.com/users/11481/Nye
Unit 8 Describing people the instructions reads Warm-up 1
Give each pair of students an envelope containing a set of picture cards and ask them to spread them out in front of them.
Choose a picture and pretend to be the person on the picture .
Describe yourself in five sentences without mentioning your name.
E.g. I have fair hair, not dark hair.
I am a woman.
My hair is short, not long.
I am wearing __ , etc.
After five sentences, ask students to guess who the person is.
[...] Main activity Give each student an envelope containing a set of picture cards .
Students then play in their pairs.
Student A puts all their pictures face up on the table.
Tell Student B to choose one picture from their own envelope and look at it without showing it to Student A. Student B pretends to be the person on the picture.
The phrase “... on the picture” as it is used is grammatical in its context.
It is short for on the picture card .
The author has omitted card from the sentence because it was mentioned in the instructions: a set of picture cards There are other cases when using on with picture , or image would fit in a teaching environment.
Focus students' attention on the picture.
Student B should focus on the image, and describe it to their partner.
Student A sticks the animals where they want on the picture.
Student B secretly draws a house on the image.
(Although " in the picture" would be more idiomatic)
ell.stackexchange.com/users/1694/Mari-Lou A
The normal thing is "in the picture" as In the picture we see Edinburgh Castle.
In English a picture is seen as a space; and a landscape, in fact, depicts a space and consequently you say: "In the picture we see a hilly countryside with a river and green meadows with sheep and in the background there is a little village."
In other languages a picture or painting may be seen not as space, but as an area, and "on the picture" is used in German.
ell.stackexchange.com/users/4159/rogermue
In the picture could be an idiomatic phrase.
It simply means that someone/thing was/wasn't concerned at all.
Say...
Do you think I'm responsible for their break up?
It is not so.
I was not in the picture at all.
On the other hand, on the picture means someone has drawn or there's a photo of a person and Student B has to be that person.
ell.stackexchange.com/users/3187/Maulik V
What is the difference between the above adverbial modifier of place and "IN the picture" if any?   In "Games for Vocabulary Practice" by Felicity O'Dell and Katie Head (Cambridge,2003) I have come across this phrase more than once reading the guidelines for running activities, e.g. "Student B pretends to be the person ON the picture".
Is it grammatically correct to say "ON the picture?"
ell.stackexchange.com
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ell.stackexchange.com/questions/37878
ell.stackexchange.com/users/6716/Yukatan
When describing the people or things which appear in a picture, we use "in the picture". For example:  In the picture we see Edinburgh Castle. If you were to use the phrase "on the picture", it would most likely be to describe something that was not part of the picture originally, but is now on it. On the picture' would refer to something not originally part of the picture. 'In the picture' is more idiomatic.
In the picture' is more idiomatic. On the picture' would refer to something not originally part of the picture.