Learn Danish: Han forvekslede mig med sin søster. - He mixed me up with his sister.

You would say: Han forvekslede mig ikke med sin søster. Because mig is an unstressed object pronoun, it steps in front of the central adverb (ikke). This is the pronoun shift you are learning about right now! If the object were a normal noun instead of a pronoun, it would stay behind the adverb: Han forvekslede ikke pigen med sin søster.

Why does the sentence use sin instead of hans for 'his'?

We use sin because the sister belongs to the subject of the sentence, Han (he). If the sentence used hans, it would mean he mixed you up with some other man's sister. Remember that sin, sit, and sine always point back to the subject of their own clause.

What is the infinitive of forvekslede and how does it work?

The infinitive is at forveksle (to mix up or confuse). It works just like English in that it almost always pairs with the preposition med (with): at forveksle en med en anden (to mix someone up with someone else).

Could I use the phrase blande sammen instead of forveksle here?

Yes, you could say Han blandede mig sammen med sin søster, which also means he mixed us up. However, forveksle is a very common and slightly more precise word when you specifically mistake one person's identity for another.