It's incredibly hard to tell when the voice is saying either "Mann" or "Mannen". Is this an issue with the translator, or is this just how Norwegian sounds?
The correct IPA pronunciations are
/mɑnːɛn/, but native speakers shorten or throw away the ə and prolong the n sound to differentiate it from mann where n stops rather abruptly.
You need to give yourself time in order to notice the subtle difference in pronunciation but until then, pay attention to the presence of the indefinite article. If there isn't any, you're dealing with definite singular of that particular noun (mannen).
When the ə (Wikipedia article) sound isn't enunciated clearly, two n sounds sort of merge and are prolonged.
When you're practising at home, replace it with a more defined e sound and give yourself time so your ears can get accustomed to the almost unnoticeable ə between the n sounds.
Click here for the guide on IPA symbols.
It's n (alveolar nasal consonant, [tongue touches the bone just behind your teeth]) because you don't bring your lips closer together to make an m sound (bilabial nasal consonant [closed lips suddenly open]).
Maybe since they are both nasal sounds, you get the impression that it's m instead of n.
The indefinite article may be omitted when stating professions or characteristics of a person.
Jeg er mann. - I am a man.
Han er vegetarianer. - He is (a) vegetarian.
Jeg er maler. - I am a painter.
Jeg er mann works but Jeg er høy mann (I am tall man) doesn't because it requires en before høy (adjective).
Other than when dealing with uncountable nouns (e.g. sukker [sugar], salt [salt]) and some exceptions aforementioned, you do need the indefinite article to be present to avoid sounding Tarzan-like.
Drikker is present tense of the verb å drikke (to drink) in Norwegian.
There is only one present tense in Norwegian which can be translated, depending on the context, as Present Continuous (is drinking) or Present Simple (drink[s]).
Thus, both "The man is drinking milk" and "The man drinks milk" are equally correct.