You can't take just that part of the sentence; the "Benim" is not connected to either of those words but the "olduğunu" is connected to the "şapkanın" so you need to include at least that.
Hangi şapka: which hat.
olduk: which is, that is, is (in a relative clause).
Hangi şapkanın ... olduğu: "..., which hat". The şapka gets a genitive ending and the olduk gets a possessive ending as normal for "noun's noun" combinations.
Hangi şapka benim: which hat is mine
Hangi şapkanın benim olduğu: "..., which hat is mine".
biliyorum: I know.
....olduğunu: same as ...olduğu plus buffer -n- needed between third person possessive marker and case marker, plus accusative case marker -u.
Hangi şapkanın benim olduğunu: "..., which hat is mine" (in accusative case)
Hangi şapkanın benim olduğunu biliyorum: "I know which hat is mine".
4 years late but i would say this is completely equivalent to "I know which hat is mine" which is the suggested answer above. it is probably less common though and to be fair you can't expect duo to have every possible wording of an answer. but i would not say it is strange or non-standard english
What helped me to understand this was looking at the question left when you break off the "I know":
- Hangi şapka benim(dir)? = Which hat is mine?
- Hangi şapkanın benim olduğunu biliyorum. = I know which hat is mine.
The dir and olduğunu both correspond to the verb "to be." The most literal translation of the Turkish sentence would be something like, "Which hat's mine-being I-know." Olduğunu breaks down like this:
- ol = root of olmak (to be/become)
- duğ = -(i)DIk object participle suffix
- u = -(s)I possessive suffix
- nu = -(s)I definite accusative suffix, with the n buffer that is used between suffixes
It might help to think of benim as being teamed up with the verb here, as in "benim olmak" = "to be mine".