Learn Farsi: آن سگ برادر او است.

Instead of adding "'s" to the owner, Farsi adds a short "e" sound (called the Ezafe) to the end of the possessed thing. It works like the English word "of". So, "dog of brother" is pronounced sag-e barādar. This "e" sound is usually not written in the Farsi script, but you must say it to connect the words.

How do we link all three words: "dog", "brother", and "his" (او)?

You simply create an Ezafe chain! You add the invisible "e" sound to each possessed word in the sequence. It literally translates to "dog [of] brother [of] him". When speaking, you will link them all together by saying sag-e barādar-e u.

Could آن سگ mean "That dog" in this sentence?

If you grouped آن سگ together without an Ezafe between them, it would mean "That dog". But if you did that, the sentence would mean "That dog is his brother"! Because the English translation is "That is his brother's dog", we know آن (that) stands alone as the subject, followed by the possession chain سگِ برادر (dog of brother).

How do we know او means "his" and not "her"?

We don't! Farsi pronouns are entirely gender-neutral. او means "he" or "she", so برادر او can mean "his brother" or "her brother". You just have to figure out which one it is based on the context of the conversation.

Why is است (is) at the very end of the sentence?