Min søn går på den samme skole som sin søster.

Because søn is a common-gender noun in Danish: en søn.

That means the correct possessive is min:

• min søn = my son

Compare:

• et hus → mit hus
• en søn → min søn
• plural nouns → mine

So the possessive form depends on the gender/number of the noun that follows.

Here går på means attends or goes to a school.

Even though gå often means walk or go, in expressions about school it is often used idiomatically:

• gå på skole = attend a school
• gå i skole = be in school / go to school in general

So this sentence is not about physically walking somewhere. It means that the son is a student at that school.

Both på and i can appear in school expressions, but they are used a bit differently.

A useful rule of thumb is:

• gå i skole = go to school / be in school in a general sense
• gå på en skole / på den skole = attend a particular school

Because this sentence mentions a specific school — den samme skole — på is the natural choice:

• Min søn går på den samme skole ...

Because Danish normally uses den/det/de samme for the same.

So:

• den samme skole = the same school

This works much like English the same.
Using only samme skole can be possible in some contexts, but in a sentence like this, den samme skole is the standard and most natural form.

Because when Danish uses a noun with a preposed adjective or similar word in a definite phrase, the definiteness is shown by den/det/de, and the noun stays in its basic form.

So:

• den samme skole
• den store bil
• det gamle hus

Not:

• den samme skolen

So the definite meaning is already marked by den.

Here som means as.

The pattern is:

• den samme ... som = the same ... as

So:

• den samme skole som sin søster = the same school as his sister

Danish uses end after comparatives such as:

• større end = bigger than
• ældre end = older than

But after samme, the correct word is som, not end.

Because sin is the reflexive possessive used when the possessor is the subject of the clause.

In this sentence, the subject is min søn, so sin søster refers back to my son:

• sin søster = his own sister

If you used hans søster, it would usually sound like you were talking about some other male person's sister, not reflexively about the subject.

So sin helps show that the sister belongs to or is connected with the subject of the sentence.

They agree with the thing possessed, not with the person who possesses it.

In this sentence:

• søster is common gender singular
• so the form is sin

Compare:

• sin søster = his/her own sister
• sit barn = his/her own child
• sine børn = his/her own children

So even though the owner is my son, the form depends on søster:

• søster → sin

Because a possessive word like sin already takes the article's place.

Just as in English you say:

• his sister

and not:

• the his sister

Danish does the same:

• sin søster

not:

• en sin søster
• sin søsteren

So sin already makes the noun phrase complete.

Because Danish, like English, often leaves out repeated words when the meaning is obvious.

So:

• Min søn går på den samme skole som sin søster

is a shortened, natural way of saying that the son attends the same school as his sister.

A fuller version could be something like:

• Min søn går på den samme skole som hans søster går på

But Danish often prefers the shorter version when the rest is easy to understand from context.

The Danish ø does not have an exact English equivalent.

A useful approximation is:

• something like the vowel in British English bird or sir
• but with rounded lips

So:

• søn has that rounded vowel
• søster begins with the same sound

If you know IPA, it is roughly:

• søn ≈ [sœn]
• søster begins with a similar ø/œ vowel

For English speakers, the most important thing is: make an eh/ur-type vowel while rounding your lips.