No, it's because this sentence wants to indicate that this egg belongs to the bird - it is her (own) egg. That is what sin/sit/sine are used for.
Here's an example to clarify:
Pigen har sin bog = The girl has her (own) book.
Pigen har hendes bog = The girl has her (could be her mom's) book.
Notice we used sin here because bog is an n-word. Since æg is a t-word, the sentence in this exercise used sit æg.
Also note that Sine is used for plurals (e.g. Hun læser sine aviser = she reads her (own) newspapers).
Note that the biological gender of the bird is irrelevant - it is the grammatical gender of the word bird that determines the pronouns.
This can be difficult for native speakers of a language that does not have grammatical genders, like e.g. English. But you have to get used to this if you learn e.g. Danish, French, German, Portuguese,...
An example for grammatical gender: In French a star (meaning a famous person) is "une star", which is a female word even if you are talking about a male singer.
Det (in this context) = it.
- Det er min fugl = It is my bird.
Sit = his/her/its own.
- Han har sit dyr = He has his (own) animal.
sin/sit/sine all mean the same and are used to imply the object/thing belonging to whomever we're talking about. Check out my answer above for more examples and clarification about when each it's used.
(I'm still learning Danish myself, but this is my theory)
The Danish word sit can be translated to his/her/it. I think the reason for the correction is that Duolingo is trying to help us learn the more common/proper way to say it. Especially where the gender (meaning male or female, not the gender type words that Danish has) of the bird hasn't been determined as far as we know. While it is common to just bestow a gender onto something, it's not the technically correct way to phrase it.
I hope that makes sense and is helpful! (Even though its probably not because I'm terrible at explaining things lol.)