Η φίλη μου και ο σύζυγός της πήγαν το μωρό στον παιδίατρο, γιατί έκλαιγε πολύ.

In Greek, it is very normal to use the definite article with a noun plus a possessive:

• η φίλη μου = my friend
• ο σύζυγός της = her husband

So Greek often says literally something like the friend of mine / the husband of hers, even though the natural English translation does not use the.

Also, when two nouns are joined by και (and), Greek usually repeats the article:

• η φίλη μου και ο σύζυγός της

That is the standard, natural phrasing.

They are possessive forms placed after the noun:

• μου = my
• της = her

So:

• η φίλη μου = my friend
• ο σύζυγός της = her husband

Grammatically, these are genitive clitic pronouns. A learner does not need all the technical detail at first; the important thing is that Greek often expresses possession as:

• article + noun + possessive word

Compare:

• το σπίτι μου = my house
• η μητέρα του = his mother
• οι φίλοι μας = our friends