Finding remainder of $123^{456}$ divided by 88 using Chinese Remainder Theorem

You used the inverse of $x_i$ instead of the inverse of $\frac{M}{m_i}$. So for example, $$ 9 \cdot \frac{88}{11} \cdot 5 + 1 \cdot \frac{88}{8} \cdot 1 \equiv 0 + 1 \cdot 11 \cdot 1 \equiv 3 \not \equiv 1\pmod{8}. $$ If you use the inverse of $\frac{M}{m_i}$ instead, you would have $$ x_1 \cdot \frac{88}{11} \cdot 8^{-1} + x_2 \cdot \frac{88}{8} \cdot 11^{-1} \equiv 0 + 1 \cdot 11 \cdot 11^{-1} \equiv 1\pmod{8}. $$

$y_1$ should've been the inverse of $8\pmod{11}$, not of $9\pmod{11}$, so $y_1=7$.

Similarly, $y_2$ should've been the inverse of $11\pmod 8$, not of $1\pmod 8$, so $y_2=3$.

Therefore, the result is: $9\times\frac{88}{11}\times \color{red}{7}+1\times\frac{88}{8}\times \color{red}{3}=537\equiv 9\pmod{88}$