"He is eating only meat" and "He is only eating meat" are in the present continuous verb tense. These sentences expresses the action that he is doing now.
"He eats only meat" and "He only eats meat" are in the simple present verb tense. They express that he has the habit of eating only meat.
Sorry, I don't know how the English sentences relate to the Danish sentence.
In longer, more complex English sentences, the position of "only" can change the meaning/emphasis of the sentence.