Min søster er enig med min mor om planen.

Because Danish often uses the pattern:

være enig med nogen om noget
= to agree with someone about something

So here:

• med min mor = with my mother
• om planen = about the plan

A natural way to understand the structure is:

My sister is in agreement with my mother about the plan.

Enig means in agreement or agreeing.

In English, we usually say is agreeing only in special situations, but in Danish være enig is the normal way to say to agree in the sense of having the same opinion.

So:

• Hun er enig = She agrees / She is in agreement

Danish often expresses this idea with the verb at være + an adjective:

• er = is
• enig = in agreement

So literally, er enig is is in agreement, even though natural English usually translates it as agrees.

This is very common in Danish:

• Jeg er enig = I agree
• Vi er enige = We agree

Because in Danish, when you use a possessive such as min, din, hans, vores, etc., you do not also use the definite ending.

So:

• min søster = my sister
• min mor = my mother

Not:

• min søsteren
• min moren

This is the normal rule in Danish: the possessive already makes the noun specific.

Because planen means the plan, while plan means a plan or just plan in a more general sense.

• en plan = a plan
• planen = the plan

The sentence is talking about a specific plan, so the definite form planen is used.

Here enig is used after er, so it is a predicative adjective.

With a singular common-gender noun like søster, the basic form enig is used:

• Min søster er enig

If the subject were plural, you would normally use enige:

• Mine søstre er enige

So the form in your sentence is correct because min søster is singular.

Because each noun needs its own possessive word if both belong to the speaker:

• min søster = my sister
• min mor = my mother

So Danish says:

Min søster er enig med min mor ...

just like English says:

My sister agrees with my mother ...

You cannot normally use one min to cover both nouns in this sentence.

Yes, in practice the agreement is mutual. If your sister agrees with your mother, then they share the same opinion.

But grammatically, the sentence is framed from the sister’s point of view:

Min søster er enig med min mor ...
= My sister agrees with my mother ...

If you switched the people, you would get:

Min mor er enig med min søster om planen.

That means essentially the same thing, but now the mother is the grammatical subject.

Yes. That is a very natural sentence, but it is structured differently.

• Min søster er enig med min mor om planen
  = My sister agrees with my mother about the plan

• Min søster og min mor er enige om planen
  = My sister and my mother agree about the plan

The second version presents them more equally as a pair. The original sentence focuses on my sister and says that her opinion matches my mother’s.

In this sentence, the word order is very close to English:

Min søster = subject
er = verb
enig med min mor om planen = rest of the sentence

So it follows a straightforward pattern:

Subject + verb + complement

However, Danish has the verb-second rule in many main clauses, so if another element comes first, the verb still usually stays in second position.

For example:

I dag er min søster enig med min mor om planen.
= Today my sister agrees with my mother about the plan.

So this sentence looks English-like, but the wider Danish word-order system is not always the same.

Not really. In normal Danish pronunciation, enig is pronounced roughly like EH-nee or AY-nee depending on how narrowly you describe it, and the final g is not pronounced like a hard English g in go.

So the important thing to remember is:

• enig does not sound like en-ig
• the last letter g is very weak or effectively silent in everyday pronunciation

The difficult part is ø, because English does not really have this vowel.

A useful approximation is:

• søster sounds roughly like SUR-ster, but with rounded lips on the first vowel

A better description:

• make a vowel somewhat like the one in French peu or German schön
• keep your lips rounded

So the main challenge is the ø sound, not the rest of the word.

Yes, Danish allows some movement, but the original order is the most neutral and natural:

Min søster er enig med min mor om planen.

This clearly keeps together:

• enig med min mor = agrees with my mother
• om planen = about the plan

Other orders may be possible in special contexts, but for a learner, the sentence as given is the safest model to copy.