Why is the CMRR of an op-amp defined in such a way?
The \$20\log_{10}(X/Y)\$ is how we change ratios into decibels or dB. 

Original, dB was used to compare ratios of power. Energy and power are often related to the square of some variable. \$1/2 mv^2\$ for kinetic energy or \$P=I^2 R\$ for power through a resistor. In this original usage, we would use \$10\log_{10}(X/Y)\$. Thus if \$X = 10 Y\$, it would be 10 dB and if \$X = 100 Y\$, it would be 20 dB. This was nice as we didn't have to deal with a bunch of decimal places to get decent comparisons.  

Now, eventually we started using dB for signals such as voltage. Voltage is related to the square root of power. \$P = V^2 / R\$. Now, convert voltage to power and place it in the dB equation:

$$10\log_{10}\left(\frac{V_1^2 / R}{V_2^2 / R}\right)$$

By the rules of logarithms, we can remove the power of 2 from the inside of the \$\log_{10}\$ to a multiplier of 2 in front of the \$\log_{10}\$. This is how we get \$20\log_{10}(X/Y)\$. Many contributions to signal processing came from electrical engineering and thus the \$20\log_{10}\$ stuck. 

Here are a few more links for reading.