Using 'a bit of water / wine ... ' vs 'a drop of water / wine' to describe a small quantity of liquid

Both the phrases are used commonly.

For ex: I am in such a pathetic state that even a drop of water will prove to be of some use.

I need a little bit of water to quench my thirst.

If you look at the synonyms for 'bit', you'll find 'piece' and 'segment' which are commonly used for solids.

We also use 'bit' in phrases like: 'a bit of sunshine', 'a bit of luck' etc.(for intangible ideas)

'A drop' is way smaller in quantity than 'a bit'. So check the context and use the most appropriate one.

More generally, can the word "bit" ...only be used for solids?

Certainly not. Using bit or little bit with liquids is perfectly normal.

Drop is acceptable too, but you should note that more often than not, drop is used in negative sentences. As in:

I stopped drinking. I haven't tasted a drop of whiskey in six months.

In positive sentences, bit is more natural. Drop sounds old-fashioned, like Janus noted, or humouous.

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PS - Bit can also be used with gases. See Google books results for 'a bit of air'

I think I have the answer. People say you can't have a bit of liquid, only a drop. But if you ask for a bit of water you are saying you want an amount of water. Measurements typically used for liquids do not always have to be used for liquid. You could have 100ml flour. So asking for a bit of flour could refer to wanting 10ml from my 100ml jug, just as asking for a bit of water could be 10ml out of 100ml. So as measurements are not fluids you can have a bit of them. Ergo, you can have a bit of liquid.