That is just how Hungarian does it when you're making 3rd person possessives. Az ő kabátja is his coat and Az ő kabátjuk is their coat. I don't know a good logical reason for it (but bear in mind that including the pronoun before a possessive is optional anyway, just for emphasis; the ending does all the work.)
But it would not change the meaning to something else when you use the more logical, but wrong ők?
The tipps state: "when the possessor is a pronoun, like ők, it loses its -k"
Which seems not be the case with önök? Which is the only other pronoun with a k on the end? As it was not chopped in half in another example.
I assume ön is also unharmed by irregularities?
It should be worded something like that: "only the pronoun ők loses its -k" period. All other pronouns stay intact.
Yes, this question surprised me too... I think the Hungarian is just indicating one single coat. It's a bit confusing, and we already have the ők -> ő change.
How many times do multiple people share a single coat....?
Maybe it would be better if this was replaced with a question that dealt with multiple coats, or replace the coat with an item (and adjective) that is more likely to be shared.