Interesting. The Arabic word for ‘drink’ (or, more accurately, ‘[he] drank’, the base form of Arabic verbs) is ‘sharaba’. Similarly, the English word ‘beer’ comes from Latin ‘bebere’ (to drink). It’s funny how Turkish (spoken in a Muslim country, and alcohol is prohibited in Islam) associates drinking with wine like English does with beer (similarly to a bunch of other languages).
Ok, I got really curious about your second question and so I went and looked it up. Basically, I think there probably isn't a Turkic word (at least not an old one - I'm sure the language commission dreamt up something in the 1930s).
According to Nişanyan Sözlük, the Arabic noun šarāb initially just means a drink (not necessarily an alcoholic one), and then there's another form, šarrāb, which means "heavy drinker, drunkard" and is first attested in the 11th century. When the word first comes into Turkish in the 12th-13th century, it is as şarāb and equates in meaning (and spelling) to the Arabic šarāb ("su, her çeşit içecek"). Turkish is pretty accurate in transliterating Arabic, so if it had meant "alcoholic drink" from the start I think we would have seen şarrāb. I would therefore guess that the association of şarap* specifically with wine is probably a much later one.
In any case, they did not need another word for wine at this point, since they already had mey, which as you note means "wine and/or a fermented beverage" in both Persian and Turkish and comes into Turkish at the same time as şarāb (the 12th-13th century). The only other Turkish word for wine I can find is bade, which is also Persian and is first attested in the early 15th century. Given that medieval Turks borrow not one but two terms for wine from Persian, it seems likely that there wasn't a Turkic word for it (and, I suppose, why would there be one when Turks probably did not enounter wine-growing regions before the 11th century and did not consistently rule over the major wine-growing areas of the Levant and coastal Anatolia until the 13th-14th century?).
That's actually what I thought, probably there was no wine in central Asia.
more to arabic "ş-r-b", meşrubat also exists in Turkish (although being used less and less) which means any kind of drink (especially non-alcoholic)
There is also şurup, which is like English syrup, which I believe comes from the same Arabic root. And şerbet.
According to Hungarian etymological dictionaries, the Hungarian word for wine, bor, derives from Old Turkic and the word can be found in Uyghur (the medieval language, I think, not the modern one), Kipchak, and Cuman, so there was a word for wine in older Turkic languages. Of course, wine was cultivated in the Caucasus, in Crimea, and I think for quite a long time in Iran, and there was a brisk trade in it throughout Central Asia, so I would have been surprised if they did not have such a word.
A little off-topic, but I got curious: it appears Persian "mey" and English "mead" have a common ancestor -- https://en.wiktionary.org/wiki/Appendix:Proto-Indo-European/m%C3%A9d%CA%B0u