1. Identify the oxidation states

• In $V_2O_3$: $2x + 3(-2) = 0 \,\Rightarrow\, 2x - 6 = 0 \Rightarrow x = +3$ So this is vanadium(III) oxide.

• In $CrO$: $x + (-2) = 0 \Rightarrow x = +2$ So this is chromium(II) oxide.

2. Use the trend for transition metal oxides

For transition metals, oxides in lower oxidation states are generally basic, while oxides in higher oxidation states become amphoteric or acidic.

3. Nature of $V_2O_3$

• Vanadium is in the $+3$ oxidation state.
• Lower oxidation state oxides of vanadium are generally basic.

Hence, $X = \text{basic}$

4. Nature of $CrO$

• Chromium is in the $+2$ oxidation state.
• $CrO$ is a basic oxide.

Hence, $Y = \text{basic}$

5. Match with the options

• A: $X=$ basic, $Y=$ amphoteric $\to$ incorrect
• B: $X=$ amphoteric, $Y=$ basic $\to$ incorrect
• C: $X=$ acidic, $Y=$ acidic $\to$ incorrect
• D: $X=$ basic, $Y=$ basic $\to$ correct

Final Answer

$\boxed{\text{D}}$