2. A. For any real numbers $a, b, c, d$, define the operation “ $\otimes$ ” between the ordered pairs of real numbers $(a, b)$ and $(c, d)$ as:
$$
(a, b) \otimes(c, d)=(a c+b d, a d+b c) \text {. }
$$

If for any real numbers $u, v$, we have
$$
(u, v) \otimes(x, y)=(u, v) \text {, }
$$

then $(x, y)$ is ( ).
(A) $(0,1)$
(B) $(1,0)$
(C) $(-1,0)$
(D) $(0,-1)$
2. A. B.

According to the defined operation rules, we have
$$
\left\{\begin{array} { l } 
{ u x + v y = u , } \\
{ v x + u y = v }
\end{array} \Rightarrow \left\{\begin{array}{l}
u(x-1)+v y=0, \\
v(x-1)+u y=0
\end{array}\right.\right.
$$

This holds for any real numbers $u, v$.
By the arbitrariness of real numbers $u, v$, we get $(x, y)=(1,0)$.