This sentence is quite weird. JÁ PIJU MLÉKO could be translated as both I drink and I am drinking milk. But JÁ MLÉKO PIJU. means that you do consume milk, that you do not hate it or that you are not allergic to it. It emphasizes the fact that you are a milk consumer. You would never say it to point out that it is an activity you are doing right now
It does not sound strange to this native at all. Please refer your native to https://www.czechency.org/slovnik/AKTU%C3%81LN%C3%8D%20%C4%8CLEN%C4%9AN%C3%8D%20V%C4%9ATN%C3%89
My (apparently incorrect) translation was, "Yes, I'm drinking milk.!" My back-story was the question: "You're drinking milk?!?!" and I wanted to acknowledge that -- although I don't usually drink milk -- I was indeed drinking it (shockingly!) at that time. Would my translation still be incorrect in that context? Just curious...