Why do sodium chloride and sodium nitrate impart the same color to the flame?

Because both salts have the same metal compound (sodium)

Explanation:

When doing a flame test the components you can identify are metals, as the change in the colour of the flame is produced by the emition of the exited electrons, in the last level of valence, returning to their groung state. And that phenomenom is common in metals, which have a trend to release electrons.

As the two salts have the same metallic compound (sodium), the flame colour will be the same for both.

Flame tests are useful in determing the presence of metallic ions via their distinct emission spectrum. Most anions such as the chloride or nitrate ions are not testible via flame tests. As such, since the metallic cation is sodium for both compounds, the flame test result is identical.