This strikes me as a strange sentence, though grammatically perfectly fine of course. It sounds weirder to me in English than in Turkish, but the implication of this sentence is that the subject drinks the object's drink as a way of habit. Semantically that does not make a lot sense to me... ;-)
Well, sorry Alex, I don't agree with Enomaes here. Sometimes the examples are quite obscure, which in terms of learning a language I find unhelpful. I'd consider dropping or changing this one. My students often can't get a point when they translate and cannot fathom the situation that makes the sentence useful.
Think of it in reverse. I can produce numerous weird, yet entirely valid sentences. Yet, they'll simply confuse rather than assist my learners.
Something to consider anyway.
Because it is the direct object in this sentence. Senin (yours) + ki (making senin a noun) + n (connecting ki and (y)i - the rule behind this is a bit technical and is related to possessive forms, but it is the reason why in this case it is seninkini rather than seninkiyi) + (y)i (the suffix indicating it is the direct object).
Not necessarily, Er/İr is a common suffix and there would be some situations imaginable where this example would go, even though it can be a bit complicated to do a non-contextual direct translation to English. Note also that the 'd' in the -Dİ suffix mutates into a 't' following 'ç', 'f', 'h', 'k', 'p', 's', 'ş', 't'. A good mnemonic aid for remembering these consonants is the following short sentence: Fe PaŞa ÇoK HaSTa (general F is very sick.... ;-) ). This also goes for a number of other suffixes that end with a 'd', like Dİr (güzeldir, hoştur).
This sentence would make sense in urdu and hindi. If we use walle for ki. Hum appwala drink pite hai it would make sense in that we drink your (kind of or your special formulated or your brand OF) drink Seninki ev bu evden çok buyuk +Tumharey wala ghar is ghar se badha hai+ Your house(the house associated with you) is bigger than this house. It can be a house which is not even yours, maybe you are a real estate agent and your clients is comparing the last house you showed him to another which another agent is showing him now.
I dont think ki has an equalent concept in English or it might be there in some archiac word but to understand how this ki is used we need to live the language and then its usage will come naturally. For the moment it is enough to try and figure out what it means when locaks use it. But like I said urdu and hindi speakers might have a better idea of how to use it