"It's my birthday!"
Translation:ʻO koʻu lā hānau kēia!
I think it's just the opposite and you can't use an additional determiner after "he", but you will often, but not always see a determiner after "‘o".. If you are stating an equivalence between two terms, there are some simple patterns for how to do that.
If one of the terms uses the determiner "he", then, it goes first. The "he" at the start of the sentence indicates that it's an equivalence, because otherwise there would be a verb before it. There can't be any other determiner (like "ka") because there is already the determiner "he". "He kumu ko‘u makuahine." (My mother is a teacher.)
If one of the terms is a proper noun, "ia", or "wai", then put that first and precede it with "‘o". In this case, you won't have a determiner after the "‘o" because proper nouns don't take determiners in Hawaiian. Seeing the "‘o" first, again alerts is that this will be an equivalence. "'O Ka‘iulani ke kumu." (Ka'iulani is the teacher.)
If neither of those is true, there may be some more minor rules for which order you put them in, but either order you add "‘o" to the start. In this case, a determiner might follow the "‘o" if the term being equated has a determiner. And, again, we can tell this is an equivalence by the "‘o" at the start. I believe that both "‘O ke kumu ko‘u makuahine," and "‘O ko‘u makuahine ke kumu," are equally as valid.
I'm not sure how to deal with a case where both terms use the determiner "he", I would guess that then the order doesn't matter, or other more minor preferences come into play. If one is a proper noun and the other uses the determiner "he", it seems the "he" wins out and comes first.
Note that the Hawaiian actually uses "kēia" ("this") and I'm sure Duolingo also accepts "This is my birthday" as a correct translation (and it might even accept "Today is", but I'm not sure about that). But in English when we want to tell someone that today is our Birthday we don't usually say, "This is" or "Today is" (they are both acceptable and you might hear them occassionally, but not very often), but instead we say, "It is my birthday!" or maybe, "It's my birthday today!" I think the Duolingo Hawaiian team has probably paired up these two sentences to help you realize that when the English sentence says "it", sometimes you need to use something else in Hawaiian, like "kēia".
It's actually like English in this. If you have a determiner (like "this" or "his"), then you don't use an article (like "the" or "a"). You can say "the day" or you can say "my day", but you can't say "the my day" or "my the day". Same in Hawaiian. "Lā" already has "ko‘u", so you can't use "ka".