Why Decibels are used to measure Signal to Noise Ratio?
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.


  I am puzzled because as you said SNR is a unit-less ratio, but at the
  same time we express it in dB...  If the ratio and its logarithm both
  do not have a unit, what then is the dB? ".


The phrase "the SNR is 50dB" is equivalent to "10 times the log of the ratio of the signal power to noise power equals 50."

The dB is not a dimensionful unit like a unit of length or of time, it is a dimensionless unit.

The number x is a pure number just as the number \$y = 10 \log(x) \$ is though we might say that "y is just x expressed in dB".