To express a ratio in dB, the ratio must be unit-less, since the logarithm of the ratio must be taken, so I'm not sure I understand why you're puzzled that we use dB.
dB is often used to express unit-less ratios precisely because of the properties of logarithm.
For example, multiplication becomes addition, division becomes subtraction.
Also, since the the signal my be many orders of magnitude greater than the noise, it is more convenient to express the SNR as, say, 50dB rather than 100,000.