"The men drank their own wine."
Translation:Mændene drak deres egen vin.
I'd like to know why it's not: drak deres egnE vin. Is the word egen/eget/egne some exception?
Yes, as is "anden/andet/andre". Though these are the only two instances where this happens (off the top of my head)
Isn't egen/eget/egne referred to the object?
Manden spiser hans egen kage, the man eats his own cake
Mændene spiser deres egen kage, the men eat their own cake
Manden spiser hans egne cager, the man eats his own cakes
Mændene spiser deres egne cager, the men eat their own cakes
Is it correct?
It would be "Manden spiser sit egen kage" and "Manden spiser sine egne kager", since those are the cakes of the man himself. Putting 'egen' there is a little redundant but for the purpose of the grammar lesson it's good.
Your forms of egen/egne are correct, but the asker was referring to the rule that adjectives after a determining article or pronoun are always written in plural form ("din røde bil"). 'Egen' and 'anden' are exceptions to that rule and follow the number and gender of the noun instead.
Can "sine" not be used here (or whatever form it should be in), or is that only used in the third person singular?
Sin/sit/sine is only for third person singluar, yes. For third person plural you only have deres.
Oh, this is very unexpected and interesting. In Norwegian and Swedish, the reflexive forms are used for the third person regardless of number. In restricting sin to the singular, Danish is odd one out.
It's rather like the Romance languages. The Latin reflexive possessive suus became the general third-person possessive in Romance. Spanish uses su and Portuguese uses o seu regardless of number, whereas French uses son and Italian uses il suo only for the singular.
It goes right back to Indo-European swé, which worked for any person.
Only using it for the third person singular seems to be a "central" Germanic invention. Old English used "sīn" for either number as well, but Old Dutch only for the singular persons. Modern Dutch "zijn" and Modern German "sein" are used only for masculine and neuter persons, and not exclusively as reflexive anymore. Danish seems to be stuck somewhere in the middle, then. :´)