Learn Polish: Jeden gen nie znaczy, że masz taki sam charakter jak brat. - One gene does not mean that you have exactly the same character as a brother.

The phrase taki sam charakter is the direct object of the verb masz (you have), which requires the accusative case. Because charakter is a masculine inanimate noun, its accusative form looks exactly like its nominative form. Therefore, the demonstrative taki sam also keeps its basic dictionary form to match.

How does the structure taki sam ... jak work here?

Taki sam means the same and jak means as. The demonstrative taki sam acts like an adjective, so it must agree in gender, number, and case with the noun it points to (in this case, charakter). Then, jak introduces the person or thing you are comparing it to.

Why is brat in the nominative case here?

In comparisons using jak, the case of the final noun depends on its role in the implied, hidden part of the sentence. The full thought is that you have the same character as your brother has (jak ma brat). Because the brother is the subject of that hidden verb, brat takes the nominative case.

Why does the sentence just say brat instead of twój brat (your brother)?