When you use the "X Y-da" construction with no "var" to talk about possession (Y has X) you're almost always talking about a specifically understood thing, describing where it is to be found/in whose possession it currently is. If you say "çay bende" you either mean a specific box of tea ("The special tea you ordered is at my place") or a specific instance of drinking tea (Okay, fine, you pay for dinner, but afterwards the tea we will drink is at my place.) This can translate weirdly as "have" in English quite often, so it's important to be aware of it.
As opposed to "Y-da X var" which generally is a little more abstract and transitory "Y has an/some X." This is more like how you'd say "they have tea" in the sense of hey, their teapot is on, lets go over and drink with them -- "onlarda cay var."
As FURTHER opposed to "Y-in X-i var" which shows a strong, intrinsic belonging.
So, let's use a cat as an example (did this elsewhere too).
Kedi bende. = The cat that doesn't necessarily belong to me is with me at the moment, and I am probably holding onto to it for someone else = "I have the cat!"
Bende kedi var. = There's a cat around my house but I wouldn't go so far as to say that I'm its loving adoring owner and it may actually not be mine forever. = "I have a cat."
Kedim var. = there exists a cat that intrinsically belongs to me and I am its owner and it is mine and owwww snuggly fluffy kittycat I WUV YOU. = "I have a cat!"
I agree, though Arabic is hard for other reasons. But you can tell when, more often than not, Google translate can make absolutely no sense of a Turkish text (does much better with Arabic or even Chinese), and I'm better off with a dictionary and the bit of Turkish i know.
Would it be correct to assume, in the logic of the Turkish language, that the reason for this is that because a thing has been located (per the -da construction) that it needs an article in translation? And by that logic, would you need to use "Bir çay onlarda" to make it "they have a (cup of) tea"?