E.g. decibels (dB) is a ratio of identical quantities $q_1$ and $q_2$. So the ratio's unit will be:
$[\frac{q_1}{q_2}] = \frac{[q_1]}{[q_1]} = \frac{1 [whatever]}{1 [whatever]} = 1 = 0dB$
So the units are gone.
If you express the ratio in logrithmic terms, you need to be notified about it. That's what the non-SI unit dB is doing: !nota bene, I'm a logarithmic number!
This comes from telecommunications. dB is the newer term, indicating $lg$ has been used. Before, at the times of first telephones, Neper was used (Np) to express ratios by $ln$.
You are free to use ratios on an anything. E.g. you could say:
- my shoes are 10 % larger than yours
- which is a 1.1 ratio
- which is $+.83$ dB (I leave it up to you to discover the hidden square and 10)