Be very careful with this. In the dative plural you do see Vögeln meaning birds, but Vögeln is also sometimes used in colloquial German as a noun in its own right with a meaning derived from the verb 'vögeln'. This verb has a very different meaning (it is a German slang term for having sex), is also only used in colloquial German as far as I know and should only be used with caution when you know exactly where to use it.
It's nost just an app, also a website, and a project. And how do you think the robotic voice came to be? Someone also recorded that, piece by piece of course. Recording all the examples in Duolingo with a person saying the sentences for all language courses would be an enormous work
It's not dropped, but realised as a glottal stop rather than as an alveolar stop.
You should hear a kind of "catch" in the voice between the two /n/ sounds.
It's a common sound change for a stop between two nasals though, as Wikipedia notes ( https://en.wikipedia.org/wiki/Standard_German_phonology#Sound_changes_and_mergers ), "speakers are often unaware of this".
So I hear the voice as saying "Enten" because I recognise the sound as being a typical realisation of the /t/ sound -- though it's not a typical realisation of /t/ for many English speakers. (Cockneys might feel at home, though, if they say wa'er bo'o for "water bottle" with a glottal stop for /t/.)
Vogel has /o:/ as its stressed vowel, a back mid rounded vowel, while Vögel has /ø:/ as its stressed vowel, a front mid rounded vowel.
I see you speak French; German “ö” is pretty much like French “eu”, the short version like in “neuf”, the long one as in “deux”. Vögel has a long ö.